A  BRIEF  ACCOUNT 


OF  THE 


HISTORICAL  DEVELOPMENT  OF 
PSEUDOSPHERICAL    SURFACES 

FROM    1827   TO    1887. 


Submitted  in  partial  fulfilment  of  the  bequirements  for  the  degree  of 

DocTOB  OF  Philosophy,  in  the  Faculty  of  Pure  Science, 

Columbia  University. 


By 
EMILY   CODDINGTON. 


Press  or 

The  Ne«  EflA  PaiNTiHC  Comfaix 

lancasieb,  Pa. 

1905 


*••  • •«• 


Sciences  i  i  n    ) 
Library  '-  ^■f' 


A   BRIEF   ACCOUNT   OF   THE   HISTORICAL   DEVELOPMENT  OF 
PSEUDOSPHERICAL   SURFACES   FROM   1827  TO  1887 


THE    APPLICATION    OF    ONE    PSEUDOSPHERICAL    SURFACE   UPON 
ANOTHER   AND   THE   GEOMETRY   OF   THESE   SURFACES. 

1.  The  definition  of  pseudosplierical  surfaces. 

2.  The  definition  of  total  curvature  according  to  Gauss. 

3.  The  application  of  surfaces  with  constant  curvature  upon  one  another. 

4.  The  pseudospherical  surfaces  of  rotation  and  the  helicoidal  surface. 

5.  Enneper's  surfaces. 

6.  Asymptotic  lines  on  a  pseudospherical  surface. 

7.  The  identification  of  the  non-Euclidean  geometry  with  pseudospherical  geometry. 

8.  The  projections  of  a  pseudospherical  surface  upon  a  plane  analogous  to  the  central 

and  stereographic  projection  of  a  sphere  on  a  plane. 

9.  The  identification  of  pseudospherical  geometry  with  the   metrical  geometry  of 

Cayley. 

II. 

THE    SURFACE    OF    CENTERS    AND    THE    TRANSFORBIATION    OF    ONE 
PSEUDOSPHERICAL  SURFACE  INTO   ANOTHER. 

1.  The  theorem  of  transformation. 

2.  Kuminer's  theory  of  congruences. 

3.  Weingarten's  two  theorems  on  surfaces  whose  radii  of  curvature  are  functionally 

related. 

4.  Ribaucour's  cyclic  system  of  surfaces. 

5.  Bianchi's  complementary  transformation. 

6.  Geodesic  lines  on  the  surface  of  centers. 

7.  Lie's  transformation. 

8.  Backlund's  transformation. 

9.  Darboux's  equations  for  Bianchi's  and  Backlund's  transformation. 

10.  The  triply  orthogonal  system  of  surfaces. 

BOOKS   OP   REFERENCE. 

1.  Dupin,  C,  Developpements  de  geometric.     Paris,  1821. 

2.  Gauss,  J.,  Disquisitiones  generales  circa  surperficies  curvas.     1827.     Translated 

into  English  by  Messrs.  Morehead  and  Hiltebeitel. 

3.  Minding,  E.  F.  A.,  Bemerkung  iiber  die   Abwickelung  krummer  Linien   von 

Fliichen.     J.  fiir  Math.,  VI.,  1830. 

4.  Lobatschewsky,  Kasaner  Boton,  1829.    Neue  Anfangsgrunde  der  Geometric  nebst 

eiuer  voUstandigen  Theorie  der  Parallelen.     Kasan,  1836-1836.     Translated 
into  French  by  J.  Hoiiel. 

1 


2  E.    M.    CODDINGTON. 

5.  Minding,  E.  F.  A.,  Ueber  die  Biegung  gewisser  Flacheu.     J.  fiir  Math.,  XVIII., 

1838. 

6.  Wie  sich  entscheiden  liisst,  ob  zwei  gegebeue  krumme  Flachen  auf  ein- 

auder  abwiekelbar  sind  oder  nicht ;  nebst  Bemerkuugen  iiber  die  Flachen 
von  unverauderlichem  Kriimmungsmasse.     J.  fvir  Math.,  XIX.,  1839. 

7.  Zur  Theorie  der  kiirzesteu  Linieu  auf  krummen  Flacheu.     J.  fiir  Math., 

XX.,  1840. 

8.  Serret,  J.  A.,  Note  sur  une  equation  aux  deriv6es  partielles.     Jouin.  de  Math., 

XIII.,  1848. 

9.  Bonnet,  0.,  M6moire  sur  la  theorie  generals  des  surfaces.     J.  de  TEc.  Pol.  cah., 

XXXII.,  1848. 

10.  Liouville,  J.,  lA".  Xote  to  Application  de  I'analyse  a  la  geometrie,  by  G.  Monge. 

1850. 

11.  Liouville,  J.,  Sur  I'equatiou  aux  diflerences  partielles,     .^  f     =h  — ^  =  0  .     Jouru. 

de  Math.,  XVIII.,  1853. 

12.  Jellet,  J.  H.,  On  the  Properties  of  the  Inextensible  Surfaces.    Transactions  of  the 

Irish  Academy,  Vol.  22,  1853. 

13.  Riemann,  B.,  Ueber  die  Hypotheseu,  welche  der  Geometrie  zu  Grunde  liegen. 

Gott.   Abh.,   1867,  written  in  1854,  translated  into  English  by  W.   K.  Clif- 
ford, Nature,  Vol.  VII. 

14.  Cayley,  A.,  Sixth  Memoir  upon  Quantics.     Philosoph.  Transactions,  Baud  149, 

1859. 

15.  Codazzi,  Nota  intorno  le  superficie,  che  hanno  costante  il  prodotto  dei  due  raggi 

di  curvatura.     Annali  di  Tortolini,  T.  VIII.,  1857. 

16.  Kummer,  Allgemeine  Theorie  der  geradlinigeu  Strahlensysteme.     Journal  von 

Crelle,  57,  1860. 

17.  Weingarten,  J.,  Ueber  eine  Klasse  auf  einander  abwickelbarer  Flachen.     J.  fiir 

Math.,  LIX.,  1861. 

18.  Enneper,  A.,  Ueber  einige  Formeln  aus  der  analytischen  Geometrie  der  Flachen. 

Schlomiich  Z.,  VII.,  1862. 

19.  Bonr,   O.,   Theorie   de    la  deformation   des   surfaces.     J.    de  TEc.    Pol.    cah. 

XXXIX.,  1862. 

20.  Weingarten,  J.,  Ueber  die  Oberfliichen,  fiir  welche  einer  der  beideu  Hauptki-iim- 

muugshalbmesser  eine  Function  des  anderen  ist.     J.  fiir  Math.,  LXU. ,  1863. 

21.  Beltrami,  E.,  lutorno  ad  alcune  propriety,  delle  superficie  di  rivoluzione.     Annali 

di  Mat.,  VI.,  1864. 

22.  Dini,  U.,  Sulla  teoria  della  superficie.     Batt.  G.,  III.,  1865. 

23.  Sulla  superficie  gobbe  che  possono,  etc.     Batt.  G.,  III.,  1865. 

24.  Sulle  superficie  di  curvatura  costante,  Batt.  G.,  III.,  1865. 

25.  Des  surfaces  a  courbure  costante.     C.  R.,  LX.,  1865. 

26.  Beltrami,  E.,  Ricerche  di  analisi  applicata  alia  geometria.     Batt.  G.,  III.,  1865. 

27.  Dimostrazione  di  due  formole  del  signor  Bonnet.     Batt.  G.,  IV.,  1866. 

28.  Risoluzione  del  problema  ;  Riportare  i  punti  di  una  superficie  sopra  un  piano 

in  modo,  che  le   linee  geodetiche   vengano   rappresentate  da  linee    rette. 
Annali  di  Mat.,  VII.,  1865. 

29.  Dini,  U.,  Sopra  alcuni  punti  della  teoria  delle  superficie  applicabili.     Annali  di 

Mat.,  VII.,  1866. 

30.  Helmholtz,  Ueber  die  thatsachlichen  Grundlagen  der  Geometrie.     1866. 


HISTORICAL   DEVELOPMENT   OF    PSEUDOSPHERICAL   SURFACES.  3 

31.  Bonnet,  0.,  Memoire  sur  la  theoiie  des  surfaces  applicables  suv  line  surface  donn6e. 

J.  de  I'Ec.  Pol.  cah.,  XLII.,  1867. 

32.  Beltrami,  E.,  Saggio  d'iuterpretazione  della  geometria  noneuelidea.     Batt.  G. , 

VI.,    1S6S.     Translated    into    French    by    J.    Houel,    Annales    de    I'^cole 
Kormale,  VI. 

33.  Cliristoffel,  E.  B.,  Allgemeine  Theorie  der  geodiitischen  Dreiecke.     Berl.  Abh., 

1868. 

34.  Dini,  U.,  Sopra  alcuni  puuti  della  teoria  delle  superficie.     Mem.  della  societa 

italiaua  (3),  I.,  18G8. 

35.  Enneper,  A.,  Analytisch-geometrische  Untersuchungen.     Gott.  Nachr.,  1868. 

36.  Die  cyklischen  Flachen.     Schlomilch  Z.,  XIV.,  1869. 

37.  Beltrami,  E.,  Teoria  fondamentale  degli  spazii  di  curvatura  costante.     Annali 

di  Mat.  (2),  II.,  1869. 

38.  Dini,  U.,  Ricerche  sopra  la  teoria  delle  superficie.     Mem.  della  societa  italiana 

(3),  II.,  1869. 

39.  Sulle  superficie,  ehe  hanno  un  sistema  di  linee  di  curvatura  plane.     Ann. 

d.  Univ.  Toscana,  1869. 

40.  Enneper,  A.,  Ueber  eine  Erweiterung  des  Begriflfs  von  Parallel  tlachen.     Gott. 

Nachr.,  1870. 

41.  Ueber  asymptotische  Linien.     Gott.  Nachr.,  1870. 

42.  Untersuchungen  iiber  einige  Puukte  aus  der  allgemeinen   Theorie   der 

Flachen.     Math.  Ann.,  II.,  1870. 

43.  Ribaucour,  A.,  Sur  la  deformation  des  surfaces.     C.  R.,  LXX.,  1870. 

44.  Dini,  U.,  Sulle  superficie,  che  hanno  un  sistema  di  linee  di  curvatura  sferiche. 

Mem.  della  societa  italiana  (3),  II.,  1870. 

45.  Klein,  F.,  Ueber  die  sogenannte  Nicht-Euklidische  Geometrie.     Math.  Ann.,  IV., 

1871. 

46.  Kretschmer,  E.,  Zur  Theorie  der  Flachen  mit  ebenen  Kriimmungslinien.     Progr. 

Frankfurt  a.  O.,  1871. 

47.  Dini,  U.,  Sopra  alcune  formole  generali  della  teoria  delle  superficie.     Annali  di 

Blat.  (2),  IV.,  1871. 

48.  Beltrami,  E. ,  Sulla  superficie  di  rotazione,  che  serve  di  tipo  alle  superficie  pseu- 

dosferiche.     Batt.  G.,  X.,  1872. 

49.  Teorema  di  geometria  pseudosferica.     Batt.  G.,  X.,  1872. 

50.  Ribaucour,  A.,  Sur  les  developpees  des  surfaces.     C.  R.,  LXXIV.,  1872. 

51.  Ribaucour,  A.,  Sur  les  systemes  cycllques.     C.  R.,  LXXVL,  1873. 

52.  Enneper,   A.,  Bemerkungen  iiber  geodatische  Linien.     Schlomilch.  Z.,  XVIII., 

1873. 

53.  Escherich,  G.  v.,  Die  Geometrie  auf  den  Flachen  constanter  negativer  Kriim- 

mung.     Wien.  Ber.,  LXIX.,  1874. 

54.  Enneper,   A.,   Bemerkung  zu  den    aualytisch-geometrischen   Untersuchungen. 

Gott.  Nachr.,  1874. 

55.  Klein,  F.,  Ueber  Nicht-Euklidische  Geometrie.     Math.  Ann.,  VII.,  1874. 

56.  Simon,  P.,  Ueber  Flachen  mit  constantem  Kriimmungsmass.     Diss.  Halle,  1876. 

57.  Enneper,  A.,  Ueber  einige  Flachen  mit  constantem   Kriimmungsmass.     Gott. 

Nachr.,  1876. 

58.  Bockwoldt,   Gr.,  Ueber    die   Enneperchen   Flachen  mit    constantem   positivem 

Kriimmungsmass.     Diss.  Gottingen,  1876. 


4  E.    M.    CODDINGTON. 

59.  Hazzadakis,  J.  N.,  Ueber  einige  Eigenschaften  der  Flacheu   mit   constantem 

Kriimmungsmass.     J.  fiir  Math.,  LXXXVIII.,  1879. 

60.  Lie,  S.,  Zur  Theorie  der  Flachen  constanter  Kriimruung.     Arch,  for  Mat.  og. 

Naturv.,  TV.,  1879. 

61.  Ueber  Flachen,  deren  Kriimmungsradieu  durch  eiue  Relation  verkuiipft 

sind.     Arch,  for  Mat.  og.  Naturv.,  IV.,  1879. 

62.  Klassifikatiou  der  Flachen  nach  der  Transformatiousgruppe  ihrer  geoda- 

tischen  Curven.       Universitatsprogr.  Christiania,  1879. 

63.  Bianchi,   L.,   Ricerche  sulle  superficie  a    curvatura  costante   e   suUe  elicoidi. 

Aunali  di  Pisa,  II.,  1879. 

64.  Cayley,  A.,  On  the  correspondence  of  homographies  and  rotations.     Math.  Ann., 

XY.,  1879. 

65.  Lecornu,  Sur  I'equilibre  des  surfaces  flexibles  et  inextensibles.    J.  de  I'Ec.  Polj'., 

XXIX.,  1880. 

66.  Lenz,  Ueber  die  Enneperschen  Flachen  constanter  negativer  Krummung.     Diss. 

Gottingen,  1879. 

67.  Lie,  S.,  Sur  les  surfaces,  dont  les  rayons  de  courbure  ont  entre  eux  une  relation. 

Darboux  Bull.  (2),  IV.,  1880. 

68.  Zur  Theorie  der  Flachen  constanter  Kriimmung.     Ai-ch.  for  Mat.   og 

Naturv.,  Y.,  1880. 

69.  Bianchi,  L.,  Ueber  der  Flachen  mit  constanter  negativer  Kriimmung.     Math. 

Ann.,  XVI.,  1880. 

70.  Backlund,   A.   V.,  Zur  Theorie  der  partiellen  DifFerentialgleichungen.     Math. 

Ann.,  XVII.,  1880. 

71.  Voss,  A.,  Ueber  ein  Priucip  der  Abbildung  krummer  Oberflachen.     Math.  Ann., 

XIX.,  1881. 

72.  Nebelung,  Trigonometrie  der  Flachen  mit  constantem  Kriimmungsmass.      Pr. 

Dortmuud,  1881. 

73.  Lie,  S.,  Trausformationstheorie  der  partiellen  Differentialgleichuug. 

,=  _,,  =  (l±^!+i!)!. 

Arch,  for  Mat.  og.  Naturv.,  VI.,  1881. 

74.  Lie,  S.,  Diskussion  der  Differentialgleichuug  s  =  F{z).    Arch,  for  Mat.  og.  Naturv., 

VI.,  1881. 

75.  Backlund,  A.  V.,  Zur  Theorie  der  Fliichentransformationen.     Math.  Ann.,  XIX., 

1881. 

76.  Haas,  A.,  Vorsuch  einer  Darstellung  der  Geschichte  des  Kriimmungsmasses.    1881. 

77.  Bianchi,  L.,  Sulle  superficie  a  curvatura  costante  positiva.     Batt.  G.,  XX.,  1882. 

78.  Weingarten,  J.,  Ueber  die  Verschiebbarkeit  geodatischer  Dreiecke  in  krummen 

Flachen.     Berl.  Ber.,  1882. 

79.  Ueber  die  Eigenschaften  des  Linienelementes  der  Flachen  von  constantem 

Kriimmungsmass.     J.  fiir  Math.,  XCIV.  u.  XCV.,  1883. 

80.  Darboux,  G.,  Sur  les  surfaces,  dont  la  courbure  totale  est  constante.     C.  R., 

XCVII.,  1883. 

81.  Sur  I'^quation  aux  derivees  partielles  des  surfaces  a  courbure  constante. 

C.  R.,  XCVII.,  1883. 

82.  Mangoldt,  H.  v.,  Klassifikatiou  der  Flacheu  nach  der  Verschiebbarkeit  ihrer 

geodatischen  Dreiecke.     J.  fiir  Math.,  XCIV.,  1883. 


HISTORICAL    DEVELOPMENT   OF    PSEUDOSPHERICAL    SURFACES.  5 

83.  Bianclii,  L.,  Sopra  alcune  classi  di  sistema  tripli  ciclici  di  superficie  ortogonali. 

I?;Ut.  O.,  XXI.,  1SS3. 

84.  Lie,  S.iUntei-suchungenuberDifferentialgleichungen.    ChristianiaForh.,  XVT^II., 

1883. 

85.  Kuen,  Th.,  Ueber  Flachen  von  constantem  Kriimmungsmass.     Munch.  Ber.,  1884. 

86.  Baicklund,  A.  V.,  Om  ytor  med  koustant  uegativ  krokuing.     Lund.  Arsskr.,  1884. 

87.  Cayley,  A.,  On  the  nou-Euclidiau  plane  geometry.     Loudon  R.  Soc.  Proceed., 

XXXYIL,  1884. 

88.  Bianchi,  L.,  Sui  sistemi  tripli  ciclici  di  superficie  ortogonali.     Batt.  G.,  XXII., 

1884. 

89.  Sopra  una  classe  di  sistemi  tripli  di  superficie  ortogonali.     Annali  di  Mat. 

(2),  XIII.,  1885. 

90.  Sopra  i  sistemi  tripli  ortogonali  di  Weingarten.     Annali  di  Mat.  (2),  XIII., 

1885. 

91.  Sopra  i  sistemi  tripli  ortogonali  di  Weingarten.     Rom.  Ace.  L.  Rend.  (4), 

I.,  1885. 

92.  Volterra,  V.,  Sulla  deformazione  delle  superficie  flessibili  ed  inestendibili.     Rom. 

Ace.  L.  Rend.  (4),  I.,  1885. 

93.  Bianchi,  L.,  Aggiunte  alia  memoria  ;  "  Sopra  i  sistemi  tripli  ortogonali  di  Wein- 

garten."    Annali  di  Mat.  (2),  XIV.,  1886. 

94.  Sopra  i  sistemi  tripli  ortogonali,  che  contengono  uu  sistema  di  superficie 

pseudosferische.     Rom.  Ace.  L.  Rend.  (4),  II.,  1886. 

95.  Weingarten,   J.,  Ueber  die    Deformationen   einer  biegsamen    unausdehnbaren 

Flache.     J.  fur  Math.,  Vol.  C,  1886. 

96.  Dobriner,  H.,  Die  Flachen  constanter  Kriimmung  mit  einem  System  spharischer 

Kriimmuugslinien.     Acta  Math.,  IX.,  1886. 

97.  Lipschitz,  R.,  Ueber  die  Oberflachen,  bei  denen  die  Differenz  der  Hauptkrum- 

mungsradien  constant  ist.    Acta  Math.,  X.,  1887. 

98.  Bianchi,  L.,  Sopra  i  sistemi  doppiamente  infiniti  di  raggi.     Rom.  Ace.  L.  Rend. 

(4),  III.,  1887. 

99.  Sui  sistemi  doppiamente  infiniti  di  raggi.     Annali  di  Mat.  (2),  XV.,  1887. 

100.  Oekinghaus,  E.,  Ueber  die  Pseudosphiire.     Hoppe  Arch.  (2),  V.,  1887. 

101.  Lilienthal,  R.  v.,  Zur  Theorie  der  Kriimmungsmittelpuuktsflachen.    Math.  Ann., 

XXX.,  1887. 

102.  Bemerkung  uber  diejenigen  Flachen,  bei  denen  die  Diflferenz  der  Haupt- 

kriimmungen  constant  ist.     Acta  Math.,  XL,  1888. 

103.  Nannei,  E.,  Le  superficie  ipercicl'che.     Napoli  Rend.  (2),  II.,  1888. 

104.  Le  superficie  ipercicliche.     Batt.  G.,  XXVI.,  1888. 

105.  Marcolongo,  R.,  Sulla  rappresentazione  conforme  della  pseudofera  e  sue  applica- 

zioni.     Napoli  Rend.  (2),  II.,  1888. 

106.  Bazzaboni,  A.,   Sopra   certe   famiglie  di  superficie   di   rivoluzione  applicabili. 

Bologna  Ace.  Rend.,  1888. 

107.  Pirondini,  G.,  Studio  sulle  superficie  elicoidali.     Annali  di  Mat.  (2),  XVI.,  1888. 

108.  Weingarten,  J.,  Ueber  eine  Eigenschaft  der  Flachen,  bei  denen  der  eine  Haupt- 

kriimmungsradius  eine  Function  des  anderen  ist.     J.  fiir  Math.,  GUI.,  1888. 

109.  LiouviUe,  R.,  Sur  les  lignes  geodesiques   des  surfaces  a  courbure  constante. 

American  J.,  X.,  1888. 

110.  Voss,  A.,  L^eber  diejeuigen  Flachen,  auf  denen  3  Scharen  geodatischer  Linien  eiu 

conjugirtes  System  bllden.     Munch  Ber.,  1888. 


6  E.    M.    CODDINGTON. 

111.  Koenigs,  G.,  Sur  Ics  surfaces,  dont  le  da"  peut  etre  rameue  de  plusieurs  manieres 

au  type  de  Liouville.     C.  R.,  CIX.,  1889. 

112.  Raflfy,  L.,  Sur  im  probleme  de  la  th^orie  des  surfaces.     Darboux  Bull.,  XIII.,  u. 

C.  R.,  CVIII.,  1889. 

113.  Chini,  M.,  Sulle  superfieie  a  curvatura  media  costante.     Batt.  G.,  XXVII.,  1889. 
11-t.  Guichard,  C,  Surfaces  rapportees  a  leurs  lignes  asjinptotiques.     Ann.  de  \"kc. 

Norm.,  VI.,  1889. 
11.5.  Beina,  V.,  Sugli  oricicli  delle  superfieie  pseudosfericlie.    Rom.  Ace.  L.  Rend.  (4), 
V,.,  1889. 

116.  Razziboni,  A.,  Delle  superfieie,  sulle  quale  due  serie  di  goedetiche  formano  un 

sistema  couiugato.     Bologna  Mem.  (4),  IX.,  1889. 

117.  Sulla  rappresentazione  di  una  superfieie  su  di  un  altra  al  modo  di  Gauss. 

Batt.  G.,  XX^^I.,  1889. 

118.  Bianchi,  L.,  Ricerche  sulle  superfieie  elicoidali.     Batt.  G.,  XXVII.,  1889. 

119.  Guichard,  C,  Recherches  sur  les  surfaces  a  courbure  totale  costante.     Ann.  de 

I'Ec.  Norm.  (3),  \T;I.,  1890. 

120.  - — sur  les  surfaces,  qui  possedent  un  rfeeau  de  g^od^siques  conjugu^es.     C. 

E.,  ex.,  1890. 

121.  Padova,  E.,  Sopra  un  teorema  di  geometria  differenziale.     1st.  Lomb.  Rend., 

XXIII.,  1890. 

122.  Bianchi,  L.,  Sopra  una  classe  di  rappresentazioni  equivalenti  della  sfera  sul  piano. 

Rom.  Ace.  L.  Rend.  (4),  Y\.,  1890. 

123.  Sopra  una  nuova  classe  di  superfieie  appartenenti  a  sistemi  tripli  orto- 

gonali.     Rom.  Ace.  L.  Rend.  (4),  VI,.,  1890. 

124.  Sulle  superfieie,  le  cui   linee  assintotiche   in   un   sistema  sono  curve  a 

torsione  costante.     Rom.  Ace.  L.  Rend.  (4),  VIj.,  1890. 

125.  Sopra  alcune  nuove   classi  di  superfieie  e  di  sistemi  tripli   ortogonali. 

Anuali  di  Mat.  (2),  XVIII.,  1890. 

126.  Schwarz,  H.  A.,  Gesammelte  Abbandlungen.     II.  Bd.,  S.  363  ff.     1890. 

127.  Picard,  E.,  Th^orie  des  equations  aux  derivees  partielles.     Journ.  de  Math.  (4), 

VI.,  1890. 

128.  Ribancour,  A.,  Memoire  sur  la  theorie  generale  des  surfaces  eourbes.    Journ.  de 

Math.  (4),  VII.,  1891. 

129.  Voss,  A.,  Zur  theorie  der  Krummung  derFlachen.     Math.  Ann.,  XXXIX.,  1891. 

130.  Cosserat,  E.,  Sur  les  systemes  cycliques  et  la  deformation  des  surfaces.     C.  R., 

CXIII.,  1891. 

131.  Sur  les  sj-st^mes  conjugues  et  sur  la  deformation  des  surfaces.     C.  R  , 

CXIII.,  1891. 

132.  Bianchi,  L.,  Sui  sistemi  tripli  ortogonali,  che  eontengono  una  serie  di  superfieie 

con  un  sistema  di  liuee  di  curvatura  plane.     Annali  di  Mat.  (2),  XIX.,  1891. 

133.  Wangerin,  A.,  Ueber  die  Abwickelung  von  Rotationsflachen   mit  constantem 

negativem  Kriimmungmass  auf  einander.     Naturf.  Ges.  Halle,  1891. 

134.  HoUaender,  E.,  Ueber  aiquivalente  Abbildung.     Diss.  Halle  u.  Progr.  Miihlheim 

a.  d.  Ruhr,  1891. 

135.  Padova,  E.,  Di  alcune  classi  di  superfieie  suscettibili  di  deformazioni  iufinitesime 

speciali.     1st.  I^omb.  Rend.  (2),  XXIV.,  1891. 

136.  Backlund,  A.  V.,  Anwendung  von  Satzen  iiber  partielle  DifTereutialgleichungen 

auf  die  Theorie  der  Orthogonalsysteme.     Math.  Ann.,  XL.,  1892. 


HISTORICAL    DEVELOPMENT    OF   PSEUDOSPHERICAL    SURFACES.  7 

137.  Bianchi,  L.,  Sulla  trasfoimazioue  di  Backluiul  per  le  superficie  pseudosferiche. 

Rom.  Ace.  L.  Rend.  (5),  I^.,  1892. 

138.  Sulla  trasformazione  di  Biickluud  pei   sistemi  tripli  ortogouali  pseudo- 

sferici.     Rom.  Ace.  L.  Rend.  (-5)  I,..  1892. 

139.  Sulle  deformazioni  iufinite.sime  delle  superficie  flessibili  ed  inestendibili. 

Rom.  Ace.  L.  Rend.  (5),  I^.,  1892. 

140.  Koenings,  G-.,   Resume  d'un   memoire   sur   les   lignes  geodesiques.     Toulouse 

Auiiales,  VI.,  1892. 

141.  Cosserat,  E.,  Sur  la  deformation  infinitdsimale  et  sur  les  surfaces  assocides  de 

Bianchi.     C.  R.,  CXV.,  1893. 

142.  Sur  les  congruences  des  droites  et  sur  la  th^orie  des  surfaces.     Toulouse 

Annales,  VII.,  1893. 

143.  Guichard,  C,  Sur  les  surfaces,  dout  les  plans  principaux  sont  equi  distants  d'un 

point  fixe.     C.  R.,  CXVI.,  1893. 

144.  Waelsch,  E.,  Sur  les  surfaces  a  Element  lineaire  de  Liouville  et  les  surfaces  a 

courbure  constante.     C.  R.,  CXVI.,  1893. 
14.5.  Ueber  die  Flachen  constanter  Kriimmung.     Wien  Ber.,  CII.,  1893. 

146.  Probst,  F.,  Ueber  Flachen  mit  isogonalen  Systemeu  von  geod  tischeu  Kreisen. 

Diss.  Wiirzburg,   1893. 

147.  BiancM,  L.,  Sulla  interpretazione  geometrica  del  teorema  di  Moutard.     Rom. 

Ace.  L.  Rend.  (5),  III„.,  1894. 

148.  Applicazioni  geometriche  del  metodo  delle  approssimazioni  successive  di 

Picard.     Rom.  Ace.  L.  Rend.  (5).  III,.,  1894. 

149.  Sulle  superficie,  i  cui  piani  principali  hanno  costante  il  rapporto  delle 

distanze  da  un  punto  fisso.     Rom.  Ace.  L.  Rend.  (5),  IIIj.,  1894. 

150.  Sui  sistemi  tripli  ortogonali  di  Weingarten.     Palermo  Rend.,  VIII.,  1894. 

151.  Soler,  E.,  Sopra  una  certa  deformata  della  sfera.     Palermo  Rend.,  VIII.,  1894. 

152.  Cosserat,  E.,  Sur  des  congruences  rectilignes  et  sur  le  problfeme  de  Ribaucour. 

C.  R.,  CXVIII.,  1894. 

153.  Genty,  E.,  Sur  les  surfaces  a,  courbure  totale  constante.     S.  M.  F.  Bull.,  XXII., 

1894. 

154.  Wangerin,  A.,  Ueber  die  Abwickelung  von  Flachen  constanten  Krummungsmasses 

sowie   einiger  anderer  Flachen  auf  einander.     Festschift  d.   Univ.   Halle 
1894. 

155.  Voss,  A.,  Ueber  isometrische  Flachen.     Math.  Ann.,  XLVI.,  1895. 

156.  Filbi,  C,  Sulle  superficie  che,  da  un  doppio  sistema  di  traiettorie  isogonali  sotto 

un  augelo  costante  delle  linee  di  curvatura,  sono  divise  in  parallelogrammi 
infinitesimi  equivalentl.     Rom.  Ace.  L.  Rend.,  IVj.,  1895. 

157.  Busse,   F.,  Ueber  diejenige  punktweise  eindeutige  Beziehung  zweier  Flachen- 

stiieke  auf  einander,  bei  weleher  jedergeodatischen  Liuie  des  eiueu  eiue  Liuie 
constanter  geodatischer  Kriimmung  des  anderan  entspricht.     Berl.-Ber. 

158.  Ueber  eine  specielle  conforme  Abbildung  der  Flachen  constanten  Kriim- 

mungsmasses  auf  die  Ebene.     1896. 

159.  Darboux,  G.,  Lemons  sur  la  theorie  g^uerale  des  surfaces.     Bd.  I. -IV.,  1887-1896. 

160.  Bianchi,  L.,  Lezioui  di  geometria  differeuziale.     Gott. ,  1894. 

161.  Genty,  E.,  Sur  la  deformation  infinit^simale  des  surfaces.     Toulouse  Ann.,  IX., 

1S96. 

162.  Goursat,  E.,  Sur  les  lignes  asymptotiques.     C.  R.,  CXXII.,  1896. 


8  E.    M.    CODDINGTON. 

163.  Calinon,  A.,  Le  theoreme  de  Gauss  sur  la  courbure.     Nouv.  Ann.,  13,  1895. 

164.  Weingarten,  J.,  Sur  la  deformation  des  surfaces.     Jour,  de  Math.  (5),  1896. 

165.  Bianchi,  Nuove  ricerche  sulla  superficie  pseudo  sferiehe.     Annali  di  Mat.,  24, 

1896. 

166.  Sopra  una  classe  die  superficie  collegate  alle  superficie  pseudo  sferiehe. 

Rom.  Accad.  Lin.  (5),  5,  1896. 

167.  Klein,  F.,  Lie's  Transformation.     Math.  Ann.,  50,  1897. 

168.  P.  Stackel  und  Fr.  Engel,  Gauss,  die  beideu  Bolyai  und  die  Nicht-Euklidische 

Geometric.     Math.  Ann.,  49,  1897. 

169.  Bukreiew,    Flachenelement  der  Flache  coustanter  Krummung.     Kiew   Univ. 

Nachr,  No.  7,  1897. 

170.  Voss,  A.,  Ueber  infinitesimalen  Flachen  Deformationen.    Deutsche  Math.,  Vol.  4, 

1896. 

171.  Zur  Theorie   der  infinitesimalen   Biegungdeformationen   einer    Flache. 

Miineh  Ber.,  27,  1897. 

172.  Bianchi,  L.,  Sur  deux  classes  de  surfaces  qui  engendrent  par  un  movement  heli- 

coidal  une  famille  de  Lame.     Toulous  Ann.,  11,  1898. 

173.  Darboux,  0.,  Legons  sur  les  systemes  orthogonaux  et  les  coordin^es  curvilignes. 

Paris,  1898. 

174.  Guichard,  C,  Sur  les  surfaces  S,  courbure  totale  constants.     C.  R.,  126. 

175.  Sur  les  systemes  orthogonaux  et  les  systemes  cycliques.     Ann.  de  I'Ec. 

Norn.  (3),  15,   1898. 

176.  Carda,  K.,  Zur  Geometrie  auf  Flachen  coustanter  Kriimmung.     Wein  Ber.,  107, 

1898. 

177.  Metzler,  G.  F.,  Surfaces  of  rotation  with  constant  measure  of  curvature.     Am.  J. 

of  Math.,  20,  1898. 

178.  Tzitzeia,  Sur  les  surfaces  a  courbure  totale  constante.     C.  R.,  128,  1898. 

179.  Darboux,  G.,  Sur  la  transformation  de  M.  Lie  et  les  surfaces  envelopp6es  de 

spheres.     Darboux  Bull.  (2),  1897. 

180.  Kommerell,V.,Bemerkungzurdenasymptotenlinien.     Boklen  Math.  (2),  2,  1900. 

181.  Waelsch,  E.,  Ueber  Flachen  mit  spharischen  oder  ebenen  Kriimmungslinien. 

Hoheschule  Briinn,  133,  1899. 

182.  Hilbert,  D.,  Ueber  Flachen  von  constanter  Gaussicher  Krummung.     Trans.  Am. 

Math.  Soc.  (2),  1,  1901. 

183.  Zuhlke,  P.,  Ueber  die  geodatischen  liuieu  und  Dreiecke  auf  den  Flachen  Con- 

stanten  Kriimmungsmasses  und  ihre  Beziehung  zur  die  sogenannten  nicht- 
Euclidischen  geometrie.     Charlottenburg,  1902. 

Note.  The  above  list  of  books  is  based  upon  the  one  given  by  Busse  at  the  end  of 
his  essay  entitled  "  Ueher  eine  specielle  conforme  Abbildung  der  Flachen  constanten 
Kriimmungsmasses  auf  die  £6ene."'^' 

In  the  following  pages  the  notation  of  the  various  authors  quoted  has  been  trans- 
lated into  the  notation  used  by  Bianchi  in  his  book  entitled,  Lezioni  di  geometria  differ- 
enziale,^^  since  the  difference  in  the  notation  used  by  the  different  writers  is  not  of 
special  interest. 

A  list  of  some  of  Bianchi's  formulae  is  added  at  the  end  of  this  paper. 


HISTORICAL    DEVELOPMENT    OF    PSEUDOSPHERICAL    SURFACES. 


THE    APPLICATION    OF    ONE    PSEUDOSPHERICAL    SURFACE 
UPON   ANOTHER   AND   THE   GEOMETRY   OF    THE 

SURFACES. 

1.  Surfaces  whose  measure  of  curvature  at  every  point  is  constant  and  nega- 
tive were  called  pseudospherical  surfaces  by  Beltrami  in  1868,  in  order,  as  he 
said,  "to  avoid  circumlocution."  Since,  therefore,  the  definition  of  these  surfaces 
depends  upon  the  definition  of  the  measure  of  curvature  itself,  their  history 
may  be  considered  as  commencing  in  1827,  when  Gauss  ^  in  his  great  memoir 
entitled  "  Disquisitiones  generales  circa  superficies  curvas  "  establislied  the  idea 
of  curvature  as  it  is  understood  today. 

2.  In  this  famous  paper  Gauss  borrowed  from  the  astronomers  the  notion 
of  spherical  representation  and  established  a  point-to-point  correspondence 
between  a  curved  surface  and  a  sphere  of  unit  radius.  He  supposed  a  ra- 
dius of  the  sphere  to  be  drawn  parallel  to  the  assumed  positive  direction  of 
the  normal  to  the  curved  surface  at  a  point  /*,  and  the  extremity  of  the 
radius  to  be  a  point  p  corresponding  to  P  of  the  surface.  He  defined  the 
total  curvature  of  a  part  of  the  surface  enclosed  within  certain  limits  as 
the  area  of  the  figure  on  the  sphere  corresponding  to  it,  and  distinguished 
this  curvature  from  the  very  important  notion  of  the  measure  of  curvature  of 
the  surface  at  a  point,  which  is  sometimes  also  called  total  curvature.  This  last 
is  defined  as  the  quotient  of  the  total  curvature  of  the  surface  element  at  the 
])oint  by  the  area  of  the  surface  element,  or  in  other  words,  "  the  ratio  of  the 
infinitely  small  areas  that  correspond  to  one  another  on  the  curved  surface  and 
on  the  sphere."  He  remarked  further  that  "  the  position  of  a  figure  on  the 
sphere  can  be  either  similar  to  the  position  of  the  corresponding  figure  on  the 
curved  surface  or  the  inverse."  When  the  position  of  two  corresponding  fig- 
ures, the  one  on  the  surface,  the  other  on  the  sphere,  is  similar,  he  called  the 
curved  surface  a  convexo-convex  surface,  or  a  surface  with  positive  curvature. 
When  the  position  of  the  figure  is  inverse  to  that  of  the  figure  on  the  surface  he 
called  the  surface  a  concavo-convex  surface  or  a  surface  with  negative  curvature. 

Gauss  introduced  various  analytic  expressions  for  this  measure  of  curvature  at 
a  point  which  he  denoted  by  K^  among  others,  using  a  general  parametric  repre- 
sentation of  a  surface  through  the  parameters  u,  v,  he  found  that 

DB"  -  D'- 
~    EG-  F-' 

where  D,  D\  D'\  E,  F  and  G,  functions   of  «  and  v,  are  tiie  coefficients  of 


10  E.    M.    CODDINGTON. 

the  first  and  second  fundamental  differential  expressions  of  the  surface,  the  one 
for  the  square  of  the  linear  element 

ds"  =  Edu^  +  2Fdudv  +  Gdv-* 
the  other 

-  1dxdX=  Ddii?  +  2D'dudv  +  D"dv\ 

where  X,  Y,  Z  are  the  direction  cosines  of   a  normal   to  the  surface  at  a   point 
X,  y,  z. 

When  he  chose  as  a  special  system  of  parametric  lines  a  family  of  geodesic 
lines  and  their  orthogonal  trajectories,  he  showed  that  E  becomes  a  function  of 
u  alone  and  F  vanishes,  so  that  the  expression  for  the  line  element  assumes  the 

form 

dr  =  dii^  +  Gdv^, 

and  he  could  derive  for  the  curvature  a  simple  corresponding  form 

1     dWG 
VG     Su?    • 

In  particular  he  observed  that  if  the  system  is  a  geodesic  polar  system  in 
which  the  u-curves  proceed  from  a  point  and  v  is  the  angle  that  each  geodesic 
«-curve  makes  with  an  arbitrary  but  fixed  I'-eurve  and  m  is  the  arc-length  of 
each  geodesic  from  the  point,  then  G  is  a.  function  which  satisfies  the  equations 


("^'"«=»-  i^^lr'- 


and  that  if  the  u-curves  form  a  geodesic  parallel  system,  that  is,  if  the  v  geo- 
desies are  orthogonal  to  a  geodesic  curve  m  =  0  and  v  is  as  before  the  arc-length 
along  the  v  curves  from  m  =  0  and  the  arc-length  v  is  measured  on  the  curve 
u  =  0  from  some  fixed  point,  the  function  G  satisfies  the  equations 

Lastly  he  wrote 


(l+p'  +  q^f' 
where  the  surface  is  represented  by 

z  =f(x,  y), 

and  J},  q,  r,  s,  t  have  their  usual   meaning  as  the  partial  derivatives  of  z  with 
respect  to  x  and  y. 


--(I)'    -^l-|.    "-Kfc')'. 

da     du'  dv     du  du     do'  dv     dv  ' 


Bianchi'6»,  §§33,  46. 


HISTORICAL    DEVELOPMENT   OP   PSEUDOSPHERICAL    SURFACES.  11 

The  measure  of  curvature  at  a  point  as  defined  by  Gauss  has  now  been 
adopted  as  the  standard  definition  and  called  the  Gaussian  measure  of  curvature 
or  total  curvature.  Both  before  and  after  the  time  of  Gauss  various  definitions 
of  curvature  of  a  surface  had  been  advanced  by  Euler,  Meusnier,  Monge  and 
Dupin,  but  these  definitions  have  not  recommended  themselves  and  are  now 
almost  forgotten. 

Gauss  did  not  write  directly  on  the  subject  of  pseudospherical  surfaces,  but 
in  his  memoir  just  quoted  he  published  two  important  discoveries  which  were 
afterwards  easily  applied  to  the  special  case  of  these  surfaces.  To  Gauss  is 
due  the  celebrated  theorem  on  the  total  curvature,  (curvatura  integra),  of  a  geo- 
desic triangle,  for  making  use  of  the  geodesic  polar  system  he  found  that  the 
total  curvature  of  a  triangle  whose  angles  are  A,  B ,  C  is 

A  +  B+  C'-TT 

which  is  negative  for  surfaces  of  negative  curvature  and  positive  for  surfaces  of 
positive  curvature.  An  immediate  inference  from  this  and  what  may  be 
regarded  as  the  first  theorem  in  the  geometry  of  pseudospherical  surfaces  is 
that  the  area  of  a  geodesic  triangle  on  one  of  these  surfaces  is  proportional  to 
its  spherical  deficiency.     This  theorem  was  proved  by  Bertrami  in  1868.* 

3.  Gauss  also  established  the  well-known  theorem  that  II  is  an  invariant  of 
bending,  that  is,  any  disturbing  of  the  shape  of  the  surface  which  does  not 
involve  stretching  or  crushing,  leaves  the  value  of  ly  at  any  point  unaltei'ed. 
Thus  if  one  surface  is  applicable  upon  another  the  measure  of  curvature  at  coi'- 
responding  points  of  the  two  surfaces  is  the  same.  It  was  this  invariant  character 
of  A'  that  first  gave  interest  to  the  study  of  surfaces  of  coustant  curvature. 
Gauss  himself  made  no  study  of  them,  but  Minding'^  in  a  paper  of  1839,  of 
which  more  will  shortly  be  said,  discussed  the  sufficiency  of  Gauss'  theorem  for 
the  applicability  of  one  surface  upon  another,  and  established  that  for  surfaces 
of  constant  curvature,  and  for  these  only,  is  Gauss'  theorem  a  sufficient  as  well 
as  a  necessary  condition. 

In  a  previous  paper  in  1830  he  had  integrated  the  Gaussian  equation 

^.__    1    d'l/G 

VG    ^«'    ' 

assuming  Kto  be  constant,  and  had  obtained  the  expression  for  the  linear  ele- 
ment 

When  about  to  apply  one  surface  upon  another  he  accordingly  wrote  for  the 
expressions  for  their  linear  elements 
*  Page  37. 


12  E.    M.    CODDINGTON. 


f/s    =  (/m    +1  -7^ —   I  dv    , 


in  which  the  primes  indicate  the  elements  with  reference  to  the  second  surface. 
The  analytical  condition  of  applicability 

ds  =  ds' 
he  satisfied  by  putting 

u  =  it\         V  ^  a  +  v' 

where  a  may  have  any  value  from  zero  to  infinity.  The  first  equation  shows 
that  any  jjoint  on  the  first  surface  may  be  made  to  correspond  with  any  point  on 
the  second  surface  and  the  second  equation  that  any  geodesic  curve  on  the  first 
surface  proceeding  from  the  point  may  be  made  to  correspond  with  any  geodesic 
curve  on  the  second  surface  proceeding  from  a  corresponding  point.  Thus  the 
surfaces  are  applicable  upon  each  other  in  oo'  ways,  or  to  quote  IMinding,  "  One 
can  place  two  arbitrary  points  of  the  one  upon  two  arbitrary  points  of  the  other, 
provided  that  the  lengths  of  the  shortest  lines  upon  the  surface  between  the 
pairs  of  points  are  equal  to  each  other.'" 

Becoming  interested  in  the  study  of  surfaces  of  constant  curvature  Minding^ 
proceeded  to  determine  some  of  these  surfaces.  When  the  surface  is  assumed 
to  be  of  the  form  of  z  =f{x,  y),  the  differential  equation  of  his  problem  is 

d^z  d'z       /  c-z  Y 
dxJ^  dy^       \  dxdy  ) 

where  /r=  a  constant. 

"This  integration,"  he  said,  "has  never  been  effected  up  to  the  present  time 
except  for  A'=  0."  Changing  the  form  of  the  equation  by  writing  .r  =  r  cosyjr 
and  y  =  r  sin  yjr  in  the  place  of  x  and  y,  he  attempted  its  solution  oulj-  for  the 
special  case 

dz 

—    =z  h  ( /i  ^  a  constant ) . 

A  first  integration  gave  him 


dz  =  I>d^lr±l^l  ^r-^-,  -l-^]d>-       (^,__  j,„„,t^,„t  „i.  i^te;;ration  , 


*"^-(^^-i-p)<"-  (5=; 


For  surfaces  of  constant  negative  curvature  he  first  put  a-  =  0  in  this  equation 
of  z  and  retained  A  :  then  he  let  It  vanish  and  u"  remain.  When  the  first  condi- 
tion is  fulfilled,  he  said,  "  the  equation  for  s  represents  a  curve  which  generates 


HISTORICAL   DEVELOPMENT    OF   PSEUDOSPHERICAL   SURFACES.  13 

the  surface  in  the  same  way  as  the  straight  line  generates  the  helicoidal  surface 
namely,  by  a  revohitiou  about  the  axis  of  :i  while  at  the  same  time  all  its  points 
have  a  common  motion  parallel  with  2."  This  surface  was  afterwards  investi- 
gated by  Dini  -*•  ■'  and  called  the  Dini  helicoidal  surface.  When  the  second 
condition  exists,  Minding  found  that  the  surface  becomes  a  surface  of  rotation 
and  the  «  curves  become  its  meridian  lines.  He  discovered  three  types  of  these 
surfaces,  those  for  which  cr  has  a  positive  value,  those  for  which  it  has  a  neg- 
ative value  and  those  for  which  it  is  equal  to  zero.  It  was  probably  his 
original  expression  for  the  linear  element 


/sin  (;<i/A')Y 
V      VK       )  " 


els-  =  dir  +  I  ^^-- —  )  dv- 

or 

ds'  =  dir  +  sinh-  udv'  {k:=  —  1) 

that  suggested  to  him  the  expedient  of  putting  i-  —  sinh  cp  in  the  equation  for  z 
when  cr  is  positive,  so  that 

s  =  J  ±  (  yl  —  cr  sinh"  (^  )d<l), 

while  for  a  negative  value  for  a"  he  naturally  introduced  the  analogous  expres- 
sion 

2=r±(i/l  — -«■  cosh-  (f})d<J3, 

Finally,  when  a-  =  0 ,  he  jjut 

r  =  — i — ,  ,  z  ^  4>  —  tanh  d>. 

cosh  <p ' 

This  last  equation  for  z  shows  that  it  represents  a  tractrix  but  Minding  did  not 
call  the  curve  by  its  name,  although  he  remarked  "  that  it  approaches  the  axis  of 
rotation  asymptotically."  He  drew  pictures  of  the  meridian  curves  of  these  three 
types  and  the  plates  appear  at  the  end  of  his  article  in  Crelle's  Journal. 
In  this  way  he  obtained  two  classes  of  pseudospherical  surfaces,  —  the  helicoidal 
surfaces  and  the  three  types  of  surfaces  of  rotation. 

Knowing  that  all  the  three  forms  of  surfaces  of  rotation  of  curvature  —1 
corresponding  to  the  three  types  of  the  meridian  curves  are  deformable  into 
each  other,  he^  wished  to  bring  their  linear  element  into  the  same  form 

ds'  =  du^  +  sinh"i((:Zy". 

He  perceived  that  this  could  be  easily  accomplished  for  surfaces  of  the  first  kind 
but  that  for  those  of  the  second  or  third  kind  it  is  necessary  to  find  for  the  para- 
metric lines  a  system  of  geodesic  lines  that  go  out  from  the  same  j)oint.  He 
developed  a  general  expression  for  the  linear  element  of  a  pseudospherical 
surface 


14 


E.    M.    CODDINGTOX. 


ds-  =  (v'  -i-  h"  —  a'- )  (It-  +    ■    ,    T^ 5 

b}'  writing  first 

ds-  =  d.vr  4-  df  +  dz"  =  dr  +  r-d-^-  + 
and  then  putting 


dv\ 


hdf+(^ 


1 


\r^ 


+  a- 


^drj, 


dt  =  d^lr  +  h 


1 


)■■  +  a- 


h- 
a- 


1 


and 


h-  +  r= 


c?r, 


V  =  r-  +  o". 

The  next  year  he  worked  out  a  set  of  equations  for  transforming  this  general 
expression  for  the  linear  element  into  the  one  referred  to  a  geodesic  polar  system 

ds"  =  da-  +  sinh-  (rdd-. 

Taking  two  points  cori-esijonding  to  coordinates  (<,  r),  [t' ,  v' )  of  the  same  sys- 
tem and  denoting  the  length  of  the  geodesic  line  which  connects  them  by  a  and 
the  angle  which  this  line  makes  with  the  curve  v  =  a  constant  by  6,  he  stated 
that  the  variables  v,  t,  cr,  9  are  related  by  the  equations 


hv'  ctn  e  -h-i?ml{t-t')      cos  6*  (V  r"  -  6=)  •  [v'  +  tanho-sin^- V  (i)'"-^^)] 


/; 


h  +  v'  ctn  6  ■  tan  h{t—  t') 
and 

V  =  sinh  a  siu 
where 

and  that  these  become 


sin  6{^/v'-  —  b-)  +  v'  tanli  a- 


Vv"  —  6-  +  v'  cosh  <T 


b-  =  a--  lr%0  and  /*  =0 
V  [t  —  t')  =  sinh  a-  cos  6, 


V  =  v'  (cosh  cr  +  sinh  cr  sin  6) 
when  6  =  0. 

He  did  not  formally  state  the  conclusion  of  this  argument  that  the  two  sets 
of  coordinates  v,  t  and  cr,  6  may  be  regarded  as  lying  on  two  different  surfaces 
and  that  then  these  equations,  instead  of  changing  the  expression  for  the  linear 
element  of  one  surface  only  into  a  slightly  different  form,  will  actually  transform 
one  surface  into  another,  and  that,  if  the  first  surface  is  supjjosed  to  be  a  surface 
of  rotation  for  which  <  =  a  constant  and  w  =  a  constant  are  the  meridian  curves 
and  the  parallels  respectively,  the  surface  will  be  one  of  the  three  types  of  which 
he  spoke  in  his  earlier  paper  according  as  to  whether 

so  that  the  above  equations  will  then  transform  a  surface  of  one  of  those  three 
types  into  one  of  the  first  type. 


HISTORICAL    DEVELOPMENT    OF   PSEUDOSPHERICAL    SURFACES.  15 

Eleven  years  after  the  publication  of  Minding's  paper  on  the  applicability  of 
surfaces,  Liouville '"  in  the  fourth  note  to  his  edition  of  Monge's  work  on  the 
"  Application  of  Analysis  to  Geometry "  sought  to  solve  the  same  problem, 
whether  the  fact  that  two  surfaces  have  the  same  measure  of  curvature  at  cor- 
responding points  is  a  sufficient  as  well  as  a  necessary  condition  for  the  deforma- 
bility  of  one  into  the  other.  His  investigations  led  him  to  criteria  of  applicability 
equivalent  to  Minding's.  No  reference  is  made  to  Minding's  paper,  but  the 
methods  used  differ  from  that  writer's.  As  Liouville  had  employed  in  the  gen- 
eral discussion  curves  of  constant  measure  of  curvature  as  a  part  of  his  coordi- 
nate system  the  results  could  not  be  extended  immediately  to  surfaces  of  constant 
curvature,  he  therefore  made  a  separate  study  of  these  surfaces.  No  new 
theorems  were  developed,  but  the  work  has  special  interest  in  that  he  employed 
isothermal  and  minimal  curves  and  made  use  of  a  new  form  for  the  expression 
for  curvature.     Taking  the  linear  element  in  the  form 

ch-  =  X  (  (It  +  d/S'- )         or  ds'  =  X  (  dudv ) 

and  writing 

II  =  a  +  i/3,         V  =  a  —  i/3 

he  obtained  a  corresponding  expression  for  total  curvature 

^'- log '^  _  A  =0 

dudv        2a-        ' 

which  is  a  differential  equation  for  the  determination  of  X  for  ^constant  and 
equal  to  —  l/fr,  and  whose  resulting  integral*  is 


X  = 


-=K'~^%^)C'^^) 


where  2f  is  the  sum  of  a  function  of  u  and  a  function  of  u,  and  2t  is  the  dif- 
ference. 

His  corresponding  expression  for  the  square  of  the  linear  element  is 

which  he  reduced  to  the  still  simpler  form, 

,  ,         a^dr 
ds-  =  -:^, 5  +  rhW^ , 


*  N.  B.  LlouviLLB  first  considered  the  question  of  the  converse  of  the  theorem  of  Gauss  in  a 
paper  published  in  the  Journal  de  Ma thematiques,  Paris,  vol.  12.  The  question  was  taken 
up  and  further  developed  by  Bertrand,  Puisseux  and  Diguet  respectively  in  papers  pub- 
lished in  vol.  13  of  the  same  Journal.  Liouville  quoted  these  writers  in  the  new  edition  of 
his  paper  which  appeared  as  Note  IV  in  the  Appendix  to  Monge's  AppUcation  de  V Analysis  d 
la  Geometric,  and  in  this  note  first  developed  the  formula  for  the  pseudosphere.  In  a  third  paper, 
very  brief  in  length,  published  iu  the  Journal  de  Mathematiques,  vol  18,  he  gave  the 
complete  details  of  the  integration  of  the  equation. 


16  E.    M.    CODDINGTON. 


2ahei 
by  putting  h9  for  t  and  r  for  ..         „.. 


Finally,  writing 


..^Oi^K^X^''-^ 


and  decomposing  the  equation  into  two  real  equations,  he  reduced  it  to 

w  ^  ' 

The  same  expression  results  when,  in  addition  to  the  constant  K ,  constants  (j 

and  h  are  introduced  in  putting 

1  _  (.i-i-'^h^i 

In  this  manner  he  established  Minding's  theorem  that  surfaces  of  the  same 
curvature  are  applicable  upon  one  another  in  oo'of  ways.  The  simplest  surface 
of  constant  negative  curvature  upon  which  all  the  surfaces  of  the  same  curvature 
are  applicable  he  found  to  be  the  surface  generated  by  the  tractrix  revolving 
about  its  asymptote. 

Codazzi  '^  was  the  next  mathematician  after  LiouviUe  to  make  a  study  of 
pseudospherical  surfaces.  He  was  the  first  to  derive  the  expression  for  the  linear 
element  of  the  surface  referred  to  a  geodesic  parallel  system  in  the  form 

ds^  =  dir  +  cosh"  udv'. 

He  integrated  Gauss'  expression  for  curvature 

_  J^  dWG 
^  ~       VG     du- 
with  the  conditions  that 


[^-^.,-,  mu-i 


p 

where  p  is  the  radius  of  geodesic  curvature  of  u  =  0,  and  obtained 

l/(y  =  cosh  u sinh  u. 

P 

When  u  =  0  is  a  curve  of  constant  geodesic  curvature,  l/p=  0,  a  hypothesis 

that  he  makes  in  connection  with  surfaces  of  revolution  and  the  formula  for 

ds^  follows. 

His  chief  contribution  to  the  subject  was  to  the  pseudospherical  trigonometry. 

He  developed  Minding's  formula  for  showing  the  relation  between  the  sides  and 

angles  of  a  geodesic  triangle 

cosh  a  =  cosh  i  cosh  c  —  sinh  b  siuh  c  cos  A , 


HISTORICAL    DEVELOPMENT   OF    PSEUDOSPHERICAL   SURFACES.  17 

and  also  the  well  known  formula 

sinb  a      sinh  h     sinh  c 


sin  A       sin  B     sin  C" 

both  of  which  are  analogous  to  those  of  spherical  trigonometry. 

Choosing  for  his  coordinates  a  geodesic  polar  system  and  writing  the  expres- 
sion for  the  linear  element  in  the  form 

ds^  =  du^  +  sinh"  udv", 

he  took  for  his  triangle  the  area  bounded  by  two  geodesic  lines  going  out  from 
the  origin,  «  =  0  and  v  =  a  constant,  and  by  a  third  geodesic  line  making  an 
angle  tt  —  /3  with  t'  =  0  at  a  distance  a  from  the  origin  and  an  angle  6  with 
■y  =  a  constant.  He  then  integrated  Gauss'  equation  for  a  geodesic  line  which 
makes  an  angle  6  with  any  parametric  geodesic  line  v  =  &  constant, 

d\/G 
dd  +  -^  dv  =  0. 
cu 

By  means  of  his  equation  for  the  linear  element  and  the  equation  for  cos  6, 

du 
cos  a  =  ~-r, 
as 

he  found  that  Gauss'  equation  becomes 

ct7ih  u  du  +  ctn  6d6  =  0 

so  that  siuh  u  sin  6  =  a  constant  =  sinh  a  sin  jS.  He  was  then  able  to  transform 
the  equation  for  d6  into 

,  sinh  udu 

as  ■ 


l/cosh^  M  —  (  sinh^  a  s\v?  y8  +  1 ) 
which,  when  integrated,  gives 

cosh  M  =  cosh  a  cosh  s  —  sinh  a  sinh  s  cos  /3, 

where  u,  a  and  s  are  the  arc-lengths  that  form  the  sides  of  the  triangle  and  yS 
is  the  angle  opposite  the  side  v. 

4.  Dini  -"■  -*•  -^'  -^  was  acquainted  with  the  work  of  Minding  and  in  the  years 
1865  and  1866  he  wrote  four  papers  in  which  among  other  things  he  developed 
more  fully  the  subjects  touched  upon  by  the  earlier  mathematicians,  the  equation 
transforming  a  pseudospherical  surface  of  rotation  of  one  of  the  three  types  into 
that  of  another  and  the  determination  of  the  form  of  a  helicoidal  surface  with 
constant  negative  curvature. 

His  method  of  finding  the  linear  element  ^^  of  a  surface  of  rotation  with  con- 
stant curvature  as  it  is  set  forth  in  his  second  paper  is  perhaps  simpler  than 


18  E.    JI.    CODDINGTON. 

that  of  his  predecessors.  Taking  the  differential  equation  for  the  length  of  the 
radius  of  curvature  of  a  meridian  curve  at  a  certain  point,  and  the  one  for  the 
length  of  the  normal  between  the  same  point  on  the  curve  and  the  axis  of  the 
revolution,  he  found  the  equation  for  the  total  curvature  —  l/cr  to  he 

dz     d^z 


v<m 


which,  when  integrated,  becomes 

d^ 

dr        \).2  ^ 


where  r  is  the  radius  of  the  parallel  circle,  z  is  the  axis  of  revolution,  z  =  (j){r) 
is  the  equation  of  the  meridian  curve  and  I  is  a  constant  of  integration.  He 
gave  to  I  the  three  values  successive^ 

a-,  0,  —  a-, 

and  thus  found  three  forms  for  the  meridian  curve, 

(1) 


dz 
dr~ 

%•=  +  d^ 

1, 

dz 

a' 

1 

1 

dz 

vl      '^^ 

1, 

dr- 

V,-  -  or 

(2) 

(3) 

and  three  corresponding  forms  for  the  linear  element 

It 

(1)  ds'  =  dxr  +  d-  sinlr  -  dv" , 

(2)  ds-  =  dir  +  IS'r'"'"  dv- , 

It 

(3)  ds-  =  dir  +  a-  cosh^  -  av- , 

where  u  and  v  represent  the  parallel  and  meridian  curves  respectively.  "The 
surfaces  (1),  (2)  and  (3),'"  he  said,  "  are  the  only  surfaces  of  revolution  with  con- 
stant negative  curvature.  They  may  be  applied  one  on  the  other,  but  as  we 
shall  see  they  constitute  as  to  their  application  three  classes  of  sharply  distinct 
surfaces.  The  second  of  tliese  classes  ( Z  =  0 )  is  made  up  of  a  single  surface, 
the  surface  which  has  for  its  meridian  the  curve  with  tangents  of  constant  length. 


HISTORICAL   DEVELOPMENT   OF   PSEUDOSPHERICAL    SURFACES.  19 

as  already  considered  by  Liouville,  and  it  serves  as  a  transition  from  surfaces 
of  the  first  class  to  those  of  the  third."' 

In  the  special  case  when  or  =  a'  he  called  the  first  surface  "  the  imaginary 
sphere,"  a  surface  which  was  afterwards  studied  by  Beltrami^-  and  Cayley  *^.* 

Dini  next  took  up  the  problem  of  the  application  of  these  surfaces  upon  one 
another.  He  remarked  that  if  in  applying  one  surface  upon  a  second  the 
meridian  curves  of  the  two  coincide,  both  surfaces  must  be  of  the  same  type. 
He  then  proceeded  to  examine  the  nature  of  the  curves  on  the  deformed  surface 
into  which  the  meridian  curves  of  the  original  surface  pass  when  the  two  sur- 
faces are  of  different  types.  For  this  purpose  he  first  sought  the  equations 
which  transform  a  surface  of  the  first  or  third  type  into  a  pseudosphere. 

When  a  surface  of  the  first  type  is  applied  upon  one  of  the  second,  he  put 
j'j/a  for  sinh  (u/a),  for  by  means  of  this  substitution  the  linear  element  of  the 
first  surface  becomes 

,  ,         «'f^'-?     ,      ,  ;  , 

ds  =  -1, ,  +  r:  dv  . 

a  form  given  by  Liouville  for  the  pseudosphere,  and  the  meridian  curves  and 
the  parallel  circles  go  over  into  a  family  of  geodesies  meeting  at  a  point  and 
their  orthogonal  trajectories.  He  found  that  the  deformation  of  a  surface  of 
the  third  type  into  a  pseudosphere  is  more  complicated.  In  his  first  paper  "■''•  *" 
he  had  developed  the  important  theorem  that  a  surface  that  possesses  a  system 
of  geodesies  intersected  at  equal  distances  by  curves  each  of  which  meets  the 
geodesies  at  the  same  constant  angle,  not  a  right  angle,  and  varying  by  a  fixed 
law  for  each  curve,  is  either  a  surface  of  rotation  or  applicable  upon  one.  He 
transformed  a  sui'face  with  constant  curvature  referred  to  such  a  system  of  geo- 
desies and  their  trajectories  for  parameters  into  a  surface  of  rotation  by  two 
methods.  By  the  application  of  the  first  method  the  trajectories  of  the  geode- 
sies on  the  first  surface  pass  over  into  parallel  circles  on  the  surface  of  rotation 
and  the  family  of  curves  that  cut  the  trajectories  orthogonally  become  its  me- 
ridian curves.  He  found  not  only  the  equations  for  this  transformation  but 
also  those  for  the  reverse  operation,  "-•  f-  ''^  for  deforming  a  surface  of  rotation  in 
such  a  way  that  its  parallels  go  over  into  a  system  of  trajectories  intersecting 
equal  lengths  on  a  family  of  geodesic  lines. 

On  the  other  hand,  when  he  deformed  the  geodesic  lines  on  his  original  sur- 
face into  the  meridians  of  a  surface  of  rotation,  the  trajectories  became  loxo- 
dromes.  He  found  that  this  last  transformation  leads  to  but  one  surface  of 
rotation,  the  pseudosphere,  and  he  introduced  in  this  connection  for  its  linear 
element  the  now  well  known  expression  "■  ''■ '" 

ds-  =  (hr  -f-  e^"dv^, 

which  did  not  appear  in  any  of  the  papers  previously  discussed. 

*  P.  40. 


20  E.    M.    CODDINGTON. 

It  was  probably  this  last  result  that  suggested  to  him  "•  "■  -^^  transforming  a 
pseudospherical  surface  of  rotation  of  the  third  type  whose  linear  element  is 

11 
ds^  =  dti^  +  or  cosh"  —  dv'' 
a 

into  a  pseudosphere  whose  linear  element  is 

ds-  =  du^  +  /3V"""-rfuJ 
by  means  of  the  equations 


/ 


du  t 

+ 


a  cosh  —  sinh 
a  a 


log  sinh     = 
°  a  a 

t  =  —  i\. 

He  found  that  the  parallels  of  the  first  surface  pass  over  into  loxodromes  on 
the  pseudosphere  and  that  the  meridians  of  the  first  surface  pass  into  the  ortho- 
gonal trajectories  of  these  loxodromes.  He  proved  that  the  cosine  of  the  angle 
at  which  these  loxodromes  meet  the  meridian  curves  is  equal  to  cosh  vja,  that 
they  possess  a  constant  geodesic  curvature  and  that  their  orthogonal  trajectories 
are  geodesic  lines. 

He  remarked  that  the  intermediate  surface,  obtained  by  using  the  first  equa- 
tion of  transformation  only,  is  the  screw  surface  of  constant  negative  curvature 
whose  helices  correspond  to  the  loxodromes  of  the  pseudosphere. 

Dini''  was  very  much  interested  in  surfaces  of  this  nature,  to  which  he  devoted 
one  paper  exclusively  in  1866,  but  before  taking  up  the  examination  of  its  con- 
tents a  digression  will  here  be  made  for  the  consideratiou  of  contributions  of 
Bianchi  and  Beltrami  to  Dini's  other  theorems.  Bianchi  ^^  introduced  a  general 
method  for  transforming  a  surface  of  rotation  of  any  of  the  three  types  into  a 
pseudosphere. 

Instead  of  choosing  a  new  system  of  geodesies  on  the  original  surface  and 
then  actually  bending  it  until  these  curves  become  the  meridians  of  the  new  sur- 
face, Bianchi  followed  Liouville's  suggestion  and  used  minimal  lines  for  parame- 
ters, so  that,  when  the  surface  is  bent,  the  lines  of  reference  remain  the  same. 

Referring  both  surfaces  to  minimal  lines  he  denoted  the  linear  element  of  the 
surface  which  he  wished  to  deform  by 

and  changed  the  expression  for  the  linear  element  of  the  pseudosphere, 

*  P.  Hi. 


HISTORICAL    DEVELOPMENT   OF   PSEUDOSPHERICAL    SURFACES.  21 

-"1 

ds'^  =  (hcl  +  6''  dv- 
into 

<^^'  =  7 ov'  d'^(l0, 

(a  +  P)" 

where  —  1/^1"  is  tiae  total  curvatm-e  of  botli  surfaces. 
His  equations  of  transformation  were  then 

1  n  1 

a  = J  +  C,  o  = i  —  C     (rt,  J  and  r  are  coDstants), 

m:  +  0  ay  —  o 

and  by  writing  for  a  and  /3  their  values  in  terms  of  u  and  u  and  for  a*  and  >/ 
their  values  in  terms  of  u  and  v,  the  parallel  and  meridian  curves  of  the  surface 
to  be  deformed,  he  found  three  sets  of  equations  for  transforming  a  surface  of 
the  first,  second  and  third  type  respectively  into  a  pseudosphere  : 


(1) 


Ml 

e 


V  e"  =  a  smh  -sm  v, 

-      «  /  .  1  "  ,  ''\ 

":=  j^l  Sinn  —  cos  v  +  cosh      I  ■ 
K  \         a  a  J 


(2) 


«1                     g— M/a 
6"=  


■u,  = 


(3) 


1—^2+   CiV-"^" 


u^  =a  log  cosh  — \-  av 


V,  =  a  tanh e" 


The  first  of  these  sets  of  equations  is  the  same  as  the  one  given  by  Minding,* 
if  in  the  latter  the  point  (1,  0)  be  chosen  for  the  point  (i',  t).  The  third  set  is 
identical  with  Dini's,  for  expressed  in  the  same  notation  Dini's  equations  become 

V  =  log  tanh  u  —  log  v^,  u^  +  log  i\  =  log  sinh  u, 

and  when  one  is  subtracted  from  the  other, 

u^  =  log  cosh  u  +  V. 

The  validity  of  transforming  the  general  expression  for  the  linear  element 

ds-  =  Edit-  +  IFdudv  +  Gdv- 

*Page  14. 


22  E.    M,    CODDINGTON. 


11  to  the  form  in  terms  of  the  conjugate  complex  variables  6  and  ^, 

ds  =  7 ^1  («  =  curvature) 


^-.y 


was  investigated  by  Weingarten"  three  years  later.  He  decided  that  such  a 
transformation  is  permissible  for  surfaces  of  constant  curvature,  and  that  then 
the  reciprocal  of  the  differential  quotient  dOjcv  must  satisfy  two  partial  dif- 
feiential  equations. 

The  geometrical  interpretation  of  the  difference  in  the  value  of  the  constant 
in  the  equation  for  the  meridian  curve  of  the  three  types  of  surfaces  of  rotation 
was  clearly  determined  by  Beltrami.'-  The  early  papers  of  this  mathematician 
on  the  subject  of  pseudospherical  surfaces  were  contemporaneous  with  those  of 
Dini  and  both  appeared  side  by  side  in  the  Italian  journals  during  the  years 
1864  and  1865. 

Beltrami-"  wrote  a  long  treatise  on  the  general  theory  of  surfaces  in  which 
he  considered  in  particular  geodesic  curvature,  evolute  and  involute  surfaces  and 
differential  parameters.  He  applied  the  various  theorems  that  he  obtained  to 
the  special  case  of  pseudospherical  surfaces,  all  of  which  theorems  will  be  dis- 
cussed in  detail  in  the  chapter  on  Evolute  Surfaces  and  tlie  Transformation 
Theory.  In  the  same  year  that  this  treatise  appeared  Beltrami  -'  published  a 
paj^er  devoted  entirely  to  the  pseudosphere  in  which  he  investigated  its  geo- 
metrical jjroperties. 

He  gave  a  geometrical  proof  of  Dini's  statement  that  the  second  class  of 
pseudospherical  surfaces  of  rotation  is  composed  of  a  single  surface,  which  is 
equivalent  to  saying  that  the  pseudosphere  is  always  bent  into  a  pseudosphere, 
that  is,  it  is  identical  with  itself,  if  bent  when  the  meridians  are  retained  as 
meridians.  He  first  showed  that  the  radius  of  geodesic  curvature  of  everj' 
parallel  of  a  pseudosphere,  being  equal  to  the  length  of  the  tangent  to  the 
meridian  curve  between  its  point  of  tangencj'  and  the  axis  of  revolution,  is  a  con- 
stant R  where  —l/H'  is  the  total  curvature  of  the  surface.  Having  proved 
that  geodesic  curvature  is  an  invariant  of  bending,  he  then  observed  that,  when 
the  pseudosphere  is  so  deformed  that  its  meridian  curves  remain  meridian  curves, 
the  radius  of  a  geodesic  curvature  of  every  parallel  circle  of  the  deformed  surface 
is  JR,  that  its  meridians  are  therefore  curves  with  tangents  of  constant  length, 
and  that  they  must  therefore  be  identical  in  form  with  the  meridians  of  the 
original  surface,  since  to  one  value  for  the  tangent  length  there  corresponds  but 
one  tractrix. 

He  also  showed,  at  this  time,  that  the  area  and  volume  of  a  pseudosphere  are 
equal  to  those  of  a  sphere  with  the  same  numerical  value  of  curvature,  thus 
making  an  analogy  between  the  simplest  forms  of  surfaces  of  rotation  with  con- 
stant positive  curvature  and  constant  negative  curvature. 


HISTORICAL    DEVELOPMENT    OF    PSEUDOSPHERICAL    SURFACES.  23 

Belti-ami's  ^-  most  important  paper  in  regard  to  surfaces  with  constant  nega- 
tive curvature  was  published  in  1868,  under  the  title  of  an  Essay  on  the  Inter- 
pretation of  the  Non-Euclidean  Geometry.  Nothing  written  on  the  subject 
since  Minding's"  paper  in  1839  can  be  compared  in  importance  with  this  cele 
brated  memoir  by  Beltrami,  in  which  for  the  fii-st  time  was  made  clear  the 
relation  of  pseudospherical  geometry  to  the  general  theory  of  geometry.  The 
contents  of  Beltrami's  paper  will  be  given  later ;  here  attention  only  will  be 
called  to  the  fact  that,  although  the  greater  part  of  the  essay  is  devoted  to  the 
demonstration  of  geometrical  propositions,  yet  it  contains  a  proof  of  his  discovery 
that  the  difference  in  the  expression  for  the  linear  element  of  the  surface  of  rota- 
tion results  from  the  difference  in  the  nature  of  the  parallel  circles  chosen  for 
one  system  of  parameters,  that  the  centers  of  those  circles  may  be  real  finite 
points,  points  at  infinity  or  imaginary  points,  and  that  consequently  the  corre- 
sjjonding  surface  will  be  of  the  first,  second  or  third  form  given  by  Dini. 

To  return  to  the  development  of  the  helicoidal  surface  with  constant  negative 
curvature  it  will  be  remembered  that  Dini  -^' "'  made  a  special  study  of  this  sur- 
face. He  first  communicated  his  results  to  the  French  Academy  iu  18G5,  and 
there  stated  that  the  surface  is  generated  by  a  tractrix  moving  along  a  helix 
that  lies  on  a  cylinder.  Recalling  Bour's'"  Theorem,  that  helicoidal  surfaces  are 
applicable  upon  surfaces  of  rotation,  he  divided  them  into  two  classes,  according 
to  whether  they  are  applicable  upon  a  sphere  or  a  pseudospherical  surface.  He 
used  the  ordinary  expressions  for  a  point  on  a  helicoidal  surface, 

X  =  u  cos  V ,     ?/  =  u  sin  v,     j;  =  niv  +  <^  (  m  ) 

where  rn  multiplied  by  Stt  is  the  rise  of  the  helix  and  0(m)  is  a  function  that 
determines  the  form  of  the  generating  profile.  He  found  that  this  generating 
profile  (^  ( w  )  must  satisfy  the  differential  equation 

in  the  case  of  surfaces  of  negative  constant  curvature  —  l/cr.     This  he  reduced  to 

„.[i +(:*)= ]=»=-„., 

by  putting  Ijm'  for  1Z-. 

Since  the  left  hand  member  of  this  last  equation  is  the  square  of  the  length 
of  the  tangent  to  the  generating  curve  between  the  axis  of  revolution  and  its 
point  of  contact  and  the  right  hand  member  is  a  constant,  Dini  thus  obtained 
an  infinity  of  new  helicoidal  surfaces  of  negative  curvature  —  l/o",  each  cor- 
responding to  a  value  of  m  and  generated  by  a  tractrix  of  tangent  length 
Va^  —  rr?  moving  about  a  cylinder  and  all  developable  upon  the  same 
pseudosphere. 


24  E.    M.    CODDINGTON. 

5.  Up  to  1868  the  ouly  surfaces  of  constant  negative  curvature  that  had  been 
determined  and  studied  were  surfaces  of  rotation  and  helicoidal  surfaces.  To 
these  were  added  a  new  group  of  surfaces  by  Enneper '''' ■*"  in  1868.  In  a 
memoir  written  in  that  year  he  determined  all  the  surfaces  of  constant  curva- 
ture, one  of  whose  families  of  lines  of  curvature  is  composed  of  plane  curves  or 
of  spherical  curves.  As  he  showed,  the  presence  of  a  system  of  plane  lines  of 
curvature  on  a  surface  of  constant  curvature  requires  that  the  second  system  of 
lines  of  curvature  should  be  spherical,  and  conversely,  moreover,  the  planes  of 
the  oue  system  meet  in  a  straight  line  and  the  centers  of  the  spheres  of  the  other 
systems  lie  on  that  line.  The  surfaces  possessing  these  characteristics,  whether 
of  positive  or  negative  constant  curvature,  have  since  been  called  Enneper's 
surfaces. 

The  determination  of  surfaces  with  either  plane  or  spherical  lines  of  curva- 
ture had  already  been  attacked  by  Joachimsthal,  Bonnet  and  Serret,  and  Bonnet' 
in  particular  examined  surfaces  for  which  the  lines  of  curvature  of  the  one 
family  are  plane  and  those  of  the  other  family  are  spherical,  and  showed  that 
for  surfaces  of  rotation  "  the  lines  of  the  one  system  are  in  planes  all  of  which 
pass  through  the  same  straight  line  and  the  centres  of  the  spheres  on  which  are 
traced  the  spherical  lines  of  curvature  can  lie  on  a  right  line." 

Enneper  considered  surfaces  of  constant  curvature  not  of  rotation  and  set 

1  1 


assuming  u  and  v  as  the  parameters  of  the  lines  of  curvature  of  which  it  =  a 
constant  are  the  plane  lines  of  curvature  and  R^  and  R^  are  the  principal  radii 
of  curvature.     Putting 

1  1  1  -  ii  1        1  1  +  < 

and  employing  the  equations 

d   V^       1    dVE  j^'^  __^  ^y"^ 

dv    R^   ~R'^  ~W  '  d^tlf^^li^  ~a^ ' 

he  obtained  the  following  equation  in  t 

,  .  d-  tan-'  t      d  tan-^  t      1  —  t^      1 

and  the  equations  for  U  and  G, 

Denoting  by  a  the  angle  the  plane  u  =  a  constant  makes  with  the  surface,  since 


I 


HISTORICAL    DEVELOPMENT    OF    P8EUD08PHERICAL    SURFACES.  25 

VEG  Su  \  li,  , 

he  substituted  the  new  variable 


R.R.  d  (\/G\  r       •       ,       , 

(4)  -— —  5-  I  —^f=r-  I  =  —  etn  0-  =  a  tuuctiou  of  u  alone, 


'■  =  11 


ctn  crdu . 


He  obtained  from  (1),  (3)  and  (4)  a  differential  erjuatiou  for  t.  Integrating 
this  equation  and  substituting  the  value  thus  obtained  for  t  in  equation  (2),  he 
differentiated  the  resulting  equation  twice  with  respect  to  ii  and  arrived  at  a 
differential  equation  for  v.     The  integral  of  this  equation 

^^—  =  A  cosh  2?/,  +  B  sinh  2m,  +  C, 

contains  three  arbitrary  constants.  The  form  of  the  surface  depends  upon  the 
value  of  these  constants  and  the  relation  that  exists  between  them. 

Enneper  concluded  there  was  no  loss  of  generality,  in  putting  B  =  0 ,  and 
investigated  accordingly.  Later  Kuen  ^  showed  that  thereby  a  class  of  surfaces 
was  overlooked. 

After  considerable  reductions  in  which  the  two  equations, 


/     dv  Y 

I  jr  y-'  I    =C  —  A  cosh  2rj  {i\  =  function  of  v  alone) , 


formed  a  chief  element,  he  proved  that  the  planes  of  the  lines  of  curvature  71  = 
a  constant  meet  in  a  straight  line,  and  that  the  lines  of  curvature  r  =  a  constant 
lie  on  spheres  that  cut  the  surface  orthogonally  and  whose  centers  lie  on  the 
straight  line. 

In  choosing  the  fixed  line  as  the  axis  of  z  and  representing  by  (/>  the  angle 
between  the  intersection  of  the  plane  with  the  xy  plane  and  the  axis  of  x,  he 
defined  the  surface  by  the  following  three  equations, 

X  sin  4>  —  y  cos  ^  =  0 , 

/  .  •     , .   <14>  sin  cr 

( X  cos  d)  +  wsina))-^-  = ,  ^ r , 

V  r-ry        r;   ^^  cosh(M,  +  «,)' 

-  i  (C'  —  A  cosh  2  V, )  dv  —  r/  tanh  ( «,  +  v, )  'l'^  =      ''^ 

'J  J  ' '  •'  ^    ^        ^'  dv       \ 

where  <^  satisfies  the  equation 

/#\2      C^-A^  .  , 
Vdu)^-^^'^-'^- 


26  E.    M.    CODDINGTON. 

A  detailed  study  of  Enneper's  surfaces  was  made  by  Bockwoldt '^'  in  1874 
for  surfaces  of  constant  positive  curvature  and  by  Lenz  ^^  in  1879  for  surfaces 
of  constant  negative  curvature  in  which  the  coordinates  of  points  on  the  surface 
are  expressed  in  elliptic  functions  of  the  two  parameters. 

There  is  one  case  and  it  was  considered  by  Enneper  in  which  the  surface  is 
expressed  through  the  elementary  functions,  namely,  when  a-  is  constant  and  B 
as  before  is  zero.     He  then  took 


and  the  equation  of  the  surface  becomes 
z=  g  cos (7  tan~'  — h  V g'''  sin^  a  —  x^  —  y^ 


X 


,       .       ,      /  ff  sin  o-  +  i/  <7"  sin^  a-  —  x^  —  y'\ 

-iff  sin  a-  log  I  —, 7   ^  .   ^  2 2  )  ' 

\c/  sma-—  y  g^  sin"  a-  —  ar  —  y''  / 

an  equation  which  shows  that  the  surface  is  generated  by  a  tractrix  whose  ver- 
tex describes  a  helix  on  a  right  circular  cylinder.  This  is  Dini's  helicoidal  sur- 
face and  it  is  thus  found  to  occupy  a  special  position  in  the  Enneper's  surfaces. 

Kuen*'^  in  an  interesting  paper  in  1884  set  forth  in  a  clear  light  the  relations 
between  the  surfaces  determined  by  Enneper  and  those  which  could  be  derived 
from  the  three  surfaces  of  rotation  by  Bianchi's  method  for  deriving  one  pseudo- 
spherical  surface  from  another  when  the  geodesic  curves  with  reference  to  which 
they  are  derived  meet  at  a  point  at  infinity.* 

6.  Geodesic  lines  and  their  orthogonal  trajectories  were  the  only  curves  con- 
sidered on  pseudosj^herical  surfaces  until  1870  when  Enneper"'  began  to  write 
on  asymptotic  curves.  Since  real  asymptotic  curves  cannot  exist  except  on  sur- 
faces of  negative  curvature,  Enneper  began  his  investigations  on  the  subject  for 
these  surfaces  only,  afterwards  supposing  the  curvature  to  be  constant  as  well 
as  negative.  Asymptotic  curves  on  pseudospherical  surfaces  possess  peculiar 
properties  that  render  them  important  in  the  infinitesimal  deformation  of  a  sur- 
face and  in  the  deriving  of  a  new  pseudospherical  surface  from  one  that  is 
known.  For  the  latter  operation  it  is  important  to  know  the  expression  for  the 
linear  element  of  the  surface  referred  to  asymptotic  curves.  Enneper  found 
this  expression  by  inserting  in  the  Codazzi-Mainardi  fundamental  equations 

d/         D        \      a  /         D'        \  dv  -^      du  dv  B 

*  Page  56. 


HISTORICAL    DEVELOPMENT    OF   PSEUDOSPHERICAL    SURFACES.  27 

dn  dv  D  \        vu  da  dv  J  D 

d  /        D"       \      d  (       D'        \         ^  dv  dv       ^  du  D 


i        B"        \       d  /        Z)'         \       ^   dv      ^    dv       ^ 


^u\^EG-F-}     ^^WEG-F-I  2{EG-F-)  VEG-F' 


[EG-F-')     VEG-F'  ^  2{EG-F')  VEG-F' 

the  values  D  =  0,  D'  =  0  and  A^=  —  l/R-  =  a  constant,  which  are  the  condi- 
tions that  the  lines  u  =  a  constant  and  v  =  a  constant  shall  be  asymptotic 
curves  on  a  surface  with  curvature  —  1/i?".     He  thus  reduced  the  equations  to 

do  du  du  dv 

from  which  he  saw  that  jB"  is  a  function  of  u  alone  and  G^  is  a  function  of  v 
alone,  so  that  the  expression  for  the  linear  element  may  be  written 

ds"  =  chr  -f  2Fdudv  +  dv' 

and  the  equation  for  the  curvature  A'  becomes 

1   _  1      5^2m 

~       i?-  ~       sin  2(u  du  dv 

where  2a>  =  the  angle  between  the  asymptotic  curves  and  F  =  cos  2a). 

Enneper  discovered  the  famous  theorem  known  as  Enneper's  theorem,  that 
the  square  of  the  radius  of  torsion  of  an  asymptotic  curve  at  every  point  is  equal 
to  the  product  of  the  principal  radii  of  curvature  of  the  surface  at  that  point, 
with  the  minus  sign  placed  before  it.  In  proving  this  theorem  he  obtained  the 
two  following  equations  for  the  curvature  l/p„  and  the  torsion  l/i\  of  the 
asymptotic  curve,  v  =  a.  constant, 

1      cmXy"^)        ~dV         1  I)' 


P„  VEG-F'  '•„      VEG-F' 

the  first  of  which  shows  that  its  geodesic  curvature  is  equal  to  its  curvature, 
and  the  second  that  its  torsion  squared  is  equal  in  value  but  opposite  iu  sign  to 
the  curvature  of  the  surface,  and  is  consequently  constant  when  the  curvature  of 
the  surface  is  constant. 

Enneper  also  remarked  that  if  one  surface  is  applied  on  another,  one  sj'stem 


28  E.    M.    CODDINGTON. 

only  of  asymptotic  curves  on  the  first  surface  can  by  any  possibility  pass  over 
into  asymptotic  curves  on  the  second  surface,  as,  for  example,  the  generators  of 
a  skew  surface. 

In  the  same  year,  Dini^'  made  a  study  of  aymptotic  curves.  Supposing  the 
surface  to  be  represented  upon  a  sphere  after  the  method  of  Gauss,  he  denoted 
its  linear  element  referred  to  arbitrary  parameters  m  and  v  by 

els-  =  Bchr  +  2Fchi  dv  +  Gdv\ 

and  the  spherical  image  of  its  linear  element  by 

ds-  =  B'du-  +  IF'dudv  +  G'dv\ 

He  derived  the  Codazzi-Mainardi  equations  for  the  coefficients  E',  F',  G',  and 
for  the  coefficients  D,  D',  D",  of  the  second  fundamental  differential  expres- 
sion for  the  surface  and  introducing  the  conditions  necessary  in  order  that  the 
parametric  lines  on  the  sphere  should  represent  the  asymptotic  curves  of  a  sur- 
face of  negative  curvature  —  A^",  he  reduced  these  equations  to 

F'  G' 

dv  ou  \fi J  ou  cv\iM J 

and  to 

dv    ~     '  du 

vehen  /a  is  a  constant. 

His  expression  for  the  spherical  representation  of  the  linear  element  is  there- 
fore, 

ds'-  =  dir  +  2F'  dudv  +  dv^ 

and  for  the  linear  element  of  the  surface  itself  it  is 

dir  —  2F'dttdv  +  dv- 

ds-  =  o , 

for  in  an  earlier  paper  he  had  remarked  that  the  arc  lengths  of  asymptotic  curves 
are  always  proportional  to  the  arc  lengths  of  their  spherical  image  in  the  i-atio 
of  the  curvature  of  the  sphere  to  the  curvature  of  the  surface  and  that  the  angle 
between  two  asymptotic  curves  on  the  surface  is  the  supplement  of  the  angle 
between  the  two  lines  that  represent  them  on  the  sphere. 

Dini  was  the  first  to  observe  from  the  form  of  this  expression  that  "  asymp- 
totic curves  divide  a  surface  into  infinitely  small  lozenges."  Hazzidakis^'  went 
a  step  further  than  Dini  and  found  the  area  of  one  of  these  lozenges  to  be 
A  +  B  +  C  +  D  —  2-  where  A,  B,  C,  D  represent  its  four  angles.  He 
obtained  this  value  by  integrating  along  its  bouudarj'  the  equation  for  the  area 
of  the  quadrilateral 


HISTORICAL   DEVELOPMENT    OF   PSEUDOSPHERICAL   SURFACES.  29 

r  fsin  2eochi  dv , 
where  sin  2(»  is  given  by  the  equation  for  the  measure  of  curvature  IT, 

sin  2a)  du  dv 

Voss"'  approached  the  subject  of  asymptotic  curves  from  the  consideration  of 
equi-distant  curves.  He  gave  that  name  to  a  system  of  curves  on  a  surface  which 
form  a  net-work  of  quadrilaterals  whose  ojiposite  side  are  equal.  His  expres- 
sion for  the  linear  element  of  the  surface,  when  u  =  a  constant  and  v  =  a  con- 
stant represent  these  lines,  becomes 

ds'-  =  (hr  +  2  cos  icodit  dv  -f  dv-, 

where  2a>  =  the  obtuse  angle  of  a  quadrilateral.  To  find  the  surface  for  which 
the  equi-distant  curves  are  asymptotic  lines  he  made  the  necessary  substitutions 
in  the  Codazzi-Mainardi  equations 

Z»=0,         Z>"  =  0, 
^=1,  G  =  l, 

and  reduced  them  to  two  partial  differential  equations 

whose  common  solution  is 

D' 

^^  =  a  constant. 


i/T 

This  expression  denotes  the  measure  of  curvature  of  the  surface  with  the  nega- 
tive sign,  so  that  Voss  thus  proved  that  only  upon  surfaces  of  constant  negative 
curvature  can  a  system  of  equi-distant  curves  be  composed  of  asymptotic  lines. 
Voss  also  found  the  characteristic  equation  for  surfaces  of  constant  negative 
curvature  —  \  j  R~ 

0-2(0       sin  2a) 

dudv         R- 

by  deforming  the  meridian  and  parallel  curves  of  the  pseudosphere  into  equi- 
distant curves. 

The  equation  for  an  asymptotic  curve  on  a  pseudosphere  was  derived  by 
Beltrami^'  in  1872.  In  that  year  he  published  a  paper  whose  title,  "On  the 
Surface  of  Rotation  that  serves  as  a  Type  for  all  Pseudospherical  Surfaces," 
shows  the  nature  of  its  contents.  In  order  to  obtain  a  set  of  equations  for  the 
surface  he  called  the  axis  of  rotation  the  axis  of  s,  the  plane  of  the  maxinuim 
parallel  circle  the  x  y  plane,  the  angle  measured  on  this  plane  that  any  meridian 


30  E.    M.    CODDINGTON. 

makes  with  a  fixed  meridian  the  angle  cj)  and  the  acute  angle  that  the  tangent 
to  the  meridian  at  anj-  point  makes  with  the  axis  of  rotation  the  angle  6.  His 
equations  for  the  coordinates  x,y,zoi  a  point  on  the  surface  of  curvature  —  l/?-" 
are  then 


0' 


:  r  sin  6  cos  <^,  y  =r  sin  ^  sin  ^,  z  =  r  I  log  ctn  „  —  cos  6  y 


Every  point  on  the  surface  is  therefore  determined  by  the  value  of  <f>  and  6  at 
that  point.     The  expression  for  the  linear  element  of  the  surface  then  becomes 

ds-  =  r'  (  ctn=  6(10-  +  sin=  6  d4>'' ) 

and  may  be  transformed  into  the  usual  form 

ds- =cht^  + r^  €-""''■  dv- 

by  means  of  the  equations 

sin^=e-"'',  <t)  =  v, 

while,  if  i?,  and  7?,  denote  the  principal  radii  of  curvature, 

i?j  =  —  J-  ctn  e,  ^.,  =  r  tan  6. 

Beltrami  began  the  study  of  asymptotic  curves  from  a  very  interesting  point  of 
view.  He  considered  first  a  curve  which  he  defined  as  "  the  curve  of  intersec- 
tion of  the  surface  with  the  tangent  plane  to  it  at  the  point  ^  =  ^^^  of  the  meri- 
dian lying  in  the  xz  plane."  He  proved  with  respect  to  this  curve  that  it  has 
two  branches  going  out  from  the  point  of  tangency  (^0  =  $^)  each  of  which 
makes  an  angle  0^  with  the  tangent  to  the  meridian  curve  and  that  when  very 
small  arc  lengths  of  the  osculating  circles  of  those  two  branches  measured  from 
the  branch  point  are  revolved  about  the  axis  of  z  they  will  generate  a  surface 
differing  by  a  quality  of  the  fourth  order  only  from  the  original  surface.  The 
two  branches  of  the  curve  of  intersection,  since  their  planes  of  osculation  are  the 
tangent  plane  to  the  surface  at  the  point  0  =  0„  will  coincide  respectively  with 
the  two  asymptotic  curves  of  the  surface  going  through  that  point ;  accordingly 
Beltrami  remarked  that  if  two  asymptotic  lines  be  drawn  through  a  point  on  a 
meridian  curve  they  will  each  make  an  angle  with  the  tangent  to  the  meridian 
at  that  point  that  is  equal  to  the  angle  which  that  same  tangent  makes  with  the 
axis  of  rotation.  He  built  up  the  following  series  of  propositions  on  this 
theorem :  — 

Since  the  equation  for  the  sine  of  the  angle  yjr  which  any  curve  makes  with 
the  geodesic  meridian  curve  ^  =  ^^  is  given  by 

d\/ G    dii  rsin0d4> 

T  —  ~     gy       ds~  ds 

and  when  the  curve  is  an  asymptotic  cuive  1^  =  0,  the  arc  length  of  an  asymp- 
totic curve  is  given  by 


HISTORICAL   DEVELOPMENT   OF   PSEUDOSPHERICAL   SURFACES.  31 

ds  =  ±  rd<f) . 
The  integral  of  this  equation 

.s  =  ;■(</. -<^„) 

shows  that  the  are  length  of  the  curve  between  the  meridans  4>  =  4>  and  4>  =  (f>^ 
is  equal  to  its  orthogonal  projection  on  the  plane  of  the  maximum  geodesic 
circle  and  that  each  one  of  the  infinite  number  of  portions  into  which  the  length 
of  the  curve  is  divided  by  a  meridian  curve  is  equal  to  the  circumference  of  the 
maximum  parallel.  If  a  linear  element  of  surface  be  taken  along  an  asymptotic 
curve,  there  results  the  equation, 

ds'  =  r-  (  ctu=  Odd-  +  sin=  Bdcj}- )  =  r^  d<f>^ 
or 

:^J^de  =  d4> 

sm  p 
and  the  integral  of  this  equation, 

0 

log  tan  2  =  </>  -  </)o 

or 

sin  6  cosh  (^(f)  —  <f>^)  =  1 , 

gives  the  equation  of  an  asymptotic  curve  that  touches  the  maximum  parallel  at 
the  point  </>„. 

The  part  played  by  an  asymptotic  line  in  the  infinitesimal  deformation  of  sur- 
faces was  discovered  by  Jellet'-  in  1853,  who  gave  the  theorem  that,  when  a  sur- 
face is  deformed  infinitely  little,  one  asymptotic  curve  may  remain  unchanged. 
He  therefore  called  asymptotic  lines  "curves  of  flexion,"  and  stated  the  proposi- 
tion, "  We  can  fix  a  curve  of  flexion  without  preventing  the  deformation  of  any 
finite  portion  of  the  surface."  As  only  surfaces  of  negative  curvature  have  real 
asymptotic  curves,  they  are  the  only  surfaces  that  can  be  bent  while  a  curve  on 
them  is  unchanged.  This  theorem  was  demonstrated  by  Lecornu'^^  in  1880  and 
by  Weingarten'^  in  1886. 

In  this  connection,  Weingarten  introduced  the  idea  of  the  bending  invariant. 
He  denoted  by  ex,  ey,  ez  the  infinitesimal  increments  that  each  coordinate 
X,  y/,  2  of  a  point  receives  when  the  surface  on  which  it  lies  is  bent  infinitely 
little,  and  by  x' ,  y\  z  the  coordinates  of  the  same  point  after  the  surface  is 
bent  so  that 

x'  =  X  4-  ex,  7/'  =y  +  ey,  z  =  z  +  ez. 

Then  assuming  the  expression  for  the  linear  element  of  the  surface  to  be 

ds'  =  Bdu'  -f  2Fdu  dv  +  Gdv; 

he  defined  the  bending  invariant  <^  by  means  of  the  equation 


32  E.    M.    CODDINGTON. 

1  /^,5i    dx      -^dx    dx\ 

'P  —  ~  yEG  —  F-  \       ^M     dv~  ^  ~dv    du)  ' 

He,  moreover,  showed  that  since  the  linear  element  of  the  surface  remains 
unchanged  when  the  surface  is  bent,  and  e  is  so  small  that  when  raised  to  the 
second  power  it  may  be  neglected,  it  must  happen  that 

2cZa;  dS  =  0 , 

ov  X,  y,z  may  be  regarded  as  the  coordinates  of  a  point  on  the  second  surface 
that  corresponds  to  the  first  by  the  orthogonality  of  its  elements. 

A  second  theorem  of  bending  that  relates  to  surfaces  of  negative  curvature  is 
that  when  two  surfaces  are  associated,  that  is,  when  the  bending  invariant  of  the 
one  at  every  point  is  equal  to  the  distance  of  the  tangential  plane  at  correspond- 
ing points  of  the  other  from  the  origin,  the  total  curvature  of  one  of  the  sur- 
faces must  be  negative.  The  discovery  both  of  the  existence  of  such  pairs  of 
surfaces  and  of  the  theorem  concerning  them  is  the  work  of  Bianchi."'*  The 
associated  surface  of  pseudospherical  surfaces  have  been  studied  within  the 
past  few  yeai-s  by  Cosserat,'^"'  "'  Guichard  '"■  "'  and  Voss. 

The  use  of  asymptotic  lines  in  the  transformation  of  one  pseudospherical  sur- 
face into  another  will  be  considered  later.  It  is  necessary  here  to  turn  to  the 
development  of  the  geometry  of  lines  on  these  surfaces. 

7.  "While  the  mathematicians  of  France,  Italy  and  Germany  were  discovering 
the  various  properties  of  surfaces  of  negative  constant  curvature  and  adding 
from  time  to  time  to  the  development  of  their  theory  so  as  to  make  them  take 
an  important  position  in  the  class  of  surfaces  of  constant  curvature,  considering 
them  merely  as  a  necessary  adjunct  to  the  completion  of  the  study  of  that  class, 
another  topic  of  more  general  interest  was  attracting  the  attention  of  men  all 
over  the  world.  This  matter  was  none  other  than  the  recognition  that  Euclid's 
fifth  postulate,  equivalent  to  the  statement  that  only  one  line  parallel  to  a  fixed 
line  can  be  drawn  through  a  point,  is  not  capable  of  demonstration  from  the  pre- 
ceding hypotheses. 

Gauss  ^  among  others  gave  some  study  to  the  subject  and  recognized  in  con- 
nection with  it  the  existence  of  a  new  geometry,  which  he  caUed  the  non- 
Euclidian,  and  which  he  distinguished  from  the  Euclidian  by  the  essential  char- 
acteristic that  in  it  there  is  never  any  similitude  in  the  figures  without  equality. 

Gauss  never  published  a  complete  exposition  of  his  theories,  but  referred  to 
them  occasionally  in  various  papers  since  published  in  the  Gottingen  edition  of 
his  works  and  in  his  correspondence  with  Schumacher.  It  is  in  a  letter  to  the 
latter  that  he  gave  the  now  familiar  expression  for  the  semi-circumference  of  a 
circle  with  radius  r  in  the  non-Euclidian  geometry, 

■n-K     ^ 

-— — (e'^  e   ^)  (  JSr=  a  constant ) 


HISTORICAL   DEVELOPMENT   OF   PSEUDOSPHERICAL    SURFACES.  33 

and  remarked  that  for  the  Euclidian  geometry  K  becomes  infinitely  great,  but 
Gauss'  contributions  to  the  new  geometry  were  slight  in  comparison  to  those  of 
Lobatchewsky. 

In  1831  Lobatchewsky*  produced  a  pamphlet  on  the  theory  of  parallel  lines, 
of  which  Gauss  said  in  another  letter  to  Schumacher,  "  I  have  found  in  the  work 
of  Lobatchewsk}^  no  surface  new  to  me,  but  the  statement  is  entirely  different 
from  that  I  had  contemplated." 

In  this  pamphlet,  Lobatchewsky  set  forth  a  whole  imaginary  geometry  based 
on  the  hypothesis  that  two  lines  parallel  to  a  third  may  be  drawn  through  a 
point  and  demonstrated  a  series  of  jjropositions  analogous  to  those  of  the  Euclid- 
ian geometry.  In  1854  Riemann'^  wrote  his  renowned  Habilitationschrift  in 
which  he  introduced  for  byperspace  of  any  dimensions  the  idea  of  three  kinds  of 
constant  curvature,  positive,  zero  and  negative.  In  particular  he  considered 
two-fold  space  and  stated  that  all  surfaces  of  positive  curvature  are  developable 
upon  a  sphere  and  all  those  of  zero  curvature  upon  a  cylinder.  "  Surfaces  of 
negative  curvature,"'  he  said,  "  will  touch  the  cylinder  externally  and  be  found 
like  the  inner  position  (towards  the  axis)  of  the  surface  of  a  ring."  He  made 
the  further  statement  that  "  the  surfaces  with  positive  curvature  can  always  be 
so  formed  that  figures  may  also  be  moved  arbitrarily  about  upon  them  without 
bending,  namely  they  may  be  formed  into  sphere  surfaces  :  but  not  those  with 
negative  curvature."  He  thus  suggested  the  idea  of  a  geometry  on  surfaces  of 
constant  negative  curvature  as  opposed  to  spherical  and  plane  Euclidian  geom- 
etry. No  mathematician,  however,  connected  the  new  geometry  of  Lobatchewskj' 
with  the  geometry  of  pseudospherical  surfaces  until  Beltrami '-  wrote  his  essay 
on  the  non-Euclidian  geometry  in  which  he  showed  analytically  that  all  the 
propositions  and  theorems  of  the  new  geometry  can  be  realized  by  means  of 
figures  lying  upon  such  a  surface. 

His  method  of  proof  was  based  upon  such  a  choice  of  parameters  u  and  v  that 
a  linear  equation  between  them  represents  a  geodesic  line  on  the  surface.  Con- 
sequently the  surface  may  be  represented  upon  a  plane  in  such  a  way  that  its 
geodesic  lines  become  straight  lines.  Beltrami  -*  wrote  a  paper  in  1865  on  this 
representation  showing  that  it  is  possible  only  for  surfaces  with  constant  curva- 
ture, and  that  it  is  analogous  to  the  central  projection  of  a  sphere  together  with 
its  linear  substitutions. 

He  rej)resented  the  geodesic  lines  by 

au  +  &y  +  c  =  0 
and  the  straight  line  on  the  plane  by 

ax  -\-hii  +  c=^ 
so  that  the  equations  for  transforming  the  one  into  the  other  are 


34  E.   Jf.    CODDINGTON. 

u  =  x,         v  =  y, 

and  the  plane  and  surface  correspond  to  each  other  point  to  point. 
Since  the  general  differential  equation  of  any  geodesic  curve  is 

2{dud'-v  —  d'udv)=  2(EG  —  F') 

f/     BE  dF        d_E\  (       DG         dE  BE  d  F\ 

[F-p-iE  ^+E^\du^-{2E  ^  -G  -^-  -ZF  -^+1F^  \dirdv 
\      cu  cu  ov  )  \        cu  cu  vv  cu  ) 

(  dE       dG  dG  dF\  (       dF        BG        BG\ 

-i-2G^+E^+^F^-2F^]dudv'+[2G^-F~-G^yM 

\  Bv  ov  Bu  Bv  J  \        Bv  Bv  Bu ) 


and  in  the  case  considered  this  must  become 

du  d'v  —  dhi  dv  =:  0 , 

he  saw  that  the  coefficient  of  each  term  of  the  right-hand  member  of  the  equa- 
tion must  vanish  identically  so  that  the  reduction  furnishes  a  set  of  equations 
whose  integrals  give  for  E,  F  and  G  the  values 


(m^  ^.  ^2  +  arf  ~  {u-  +  v-  +  a')-'  ~  (?r  +  v-  +  a^f 

where  l/H-  is  the  curvature  of  the  surface  and  a  is  an  arbitrary  constant. 

He  was  thus  able  to  write  down  at  once  the  expression  for  the  linear  element 
of  a  surface  with  constant  positive  curvature 

-S-  ( ( rr  +  1)^ )  dii^  —  2uvdu  do  -f  ( a"  -f  ir )  dv'  \ 

(ir  +  V  +  a-y 
which  he  changed  into 

i?"  ((«'  —  v-)dir  +  'luvdudv  +  («-  —  u')dv- ) 
ds'  = 


for  pseudospherical  surfaces  by  wi-iting  —  H'  and  —  a'  for  S'  and  a'. 

This  expression  for  the  linear  element  was  his  starting  point  for  his  investi- 
gations on  the  non-Euclidian  geometry  in  1868  and  from  it  be  developed  other 
properties  peculiar  to  the  surface  and  to  Lobatchewsky's  imaginary  plane.  He 
observed  that  if  0  be  the  angle  between  two  lines  u  =  a  constant  and  v  =  a 
constant  that 


uv  .    ^  aVa'  —  u^  —  v' 

cos  ff  =  ,  sm  ^  = 


By  using  polar  coordinates  r  and  (f>  he  found  a  second  expression  for  the  linear 
element  for  the  surface 


HISTORICAL    DEVELOPMENT    OF   rSEUDOSPHERICAL   SURFACES.  35 


*  =  ^'|(^0'-^^=1' 


From  this  he  derived  equations  for  the  length  /o  of  a  geodesic  line  (^  =  a  con- 
stant and  for  the  arc  o-  of  a  geodesic  circle  ;■  =  a  constant,  or  as  he  called  it,  a 
geodesic  circumference, 


B  ,       a+\/v?  +  v^       R^       a+r 


)•  =  a  tauh 


p 
a  =  <^R  sinh  j^  . 

His  expression  for  the  circumference  of  a  geodesic  circle  is  therefore  similar  to 
to  the  one  found  by  Gauss, 

7r^(e'=''^-e-''>'). 

From  these  equations  he  saw  that  the  curve  whose  equation  is 

u'  +  V'  =  «', 

bounds  the  region  of  real  values.  He  remarked  that  when  the  surface  is  repre- 
sented upon  a  plane  this  curve  becomes  a  circle  which  he  called  the  limiting 
circle  because  all  the  points  corresponding  to  real  points  on  the  surface  lie 
within  it,  all  those  corresponding  to  the  ideal  or  imaginary  points  on  the  surface 
lie  without  it  and  the  points  on  its  circumference  correspond  to  infinitely  far  off 
points  on  the  surface.  He  also  showed  that  the  geodesic  lines  of  the  surface 
become  chords  of  this  circle  and  that,  since  two  points  fully  determine  a  chord, 
two  points  will  determine  a  geodesic  line. 

From  the  equation  for  d  he  further  observed  the  nature  of  the  parametric  lines 
on  the  surface,  that  they  consist  of  two  systems  of  geodesic  lines  which  ai-e  so 
related  to  each  other  that  the  two  fundamental  lines  u  =  0,  u  =  0,  meet  at  right 
angles  at  the  origin  while  the  coordinate  lines  u  =  a  constant  are  orthogonal  to 
V  =  0  and  the  coordinate  lines  v  —-  a  constant  are  orthogonal  to  ?f  =  0  . 

He  showed  by  a  rigorous  proof  that  any  two  lines  that  cut  each  other  orthog- 
onally may  be  chosen  for  the  fundamental  lines,  instead  of  m  =  0 ,  v  =  0 ,  and 
that  consequently  any  geodesic  line  may  be  made  to  coincide  with  any  other  and 
the  surface  superposed  upon  itself,  for  changing  the  pair  of  orthogonal  geo- 
desic lines  intersecting  at  the  origin  into  any  other  set  of  orthogonal  geodesic 
lines  through  any  other  point  does  not  alter  the  form  of  the  expression  for  the 
linear  element. 

These  two  characteristics,  the  superposability  of  the  surface  upon  itself  and 
the  determination  of  a  geodesic  line  by  two  points,  Beltrami  called  the  "  funda- 


36  E.    M.    CODDINGTON. 

mental  criteria  of  elementary  geometry,"  and  since  they  belong  equally  to  pseiido- 
spberieal  surfaces  and  to  the  Lobatchewskian  plane,  he  said :  "  It  becomes 
evident  that  the  theorems  of  the  plane  non-Euclidian  geometry  exist  uncon- 
ditionally for  all  the  surfaces  of  constant  negative  curvature." 

The  keystone  of  the  non-Euclidian  geometry  is  the  proposition  that  two 
straight  lines  can  be  drawn  through  any  fixed  point  parallel  to  a  given  straight 
line.  Beltrami  proved  this  proposition  by  means  of  his  geodesic  representation 
of  the  pseudospherical  surface  on  the  plane  in  the  following  manner:  first  it  is 
necessary  to  show  that  the  angle  between  two  geodesic  lines  intersecting  at  a 
real  finite  point  on  the  surface  is  never  0  nor  tt,  but  that  the  angle  may  be  0 
or  TT  when  the  curves  intersect  at  a  point  of  infinity.  If  this  angle  is  repre- 
sented by  yfr  and  the  angle  on  the  plane  between  two  chords  corresponding  to  the 
geodesic  lines  be  represented  by  ifr'  and  the  angles  which  the  chords  make  with 
the  axis  of  ^Y  by  /m  and  v  respectively,  -\]r  and  -yjr'  are  related  by  means  of  the 
equation 

a{\/a^  —  u^  —  v^)  sin  -v/r' 


tan  i/r 


'  cos  1^'  —  ( w  cos  /A  —  u  sin  /^ )  ( t'  cos  v  —  u  sin  v ) ' 


the  right-hand  member  of  which  can  only  be  zero  when  ?r  -)-  v"  =  «",  that  is 
when  the  two  chords  meet  on  the  perimeter  of  the  circle,  consequently  the  angle 
i/r  is  0  only  when  the  geodesic  curves  meet  at  a  point  at  infinity  which  corresponds 
with  the  point  of  intersection  of  the  chords  on  the  perimeter  of  the  limiting 
circle. 

When  a  given  geodesic  and  a  given  point  on  the  surface  are  represented  by  a 
chord  of  the  limiting  circle  on  the  plane  and  a  point  within  its  perimeter,  two 
of  the  chords  drawn  through  that  point  will  meet  the  first  chord  at  its  extremi- 
ties on  the  circumference  of  the  circle,  therefore,  the  two  geodesic  lines  which 
correspond  to  the  two  chords  will  meet  the  given  geodesic  line  at  infinity,  mak- 
ing with  it  an  angle  0  and  they  will  therefore  both  be  parallel  to  it  though 
drawTi  through  one  point. 

Following  the  line  of  thought  laid  down  by  Lobatchewsky,  Beltrami  next 
defined  the  angle  of  parallelism  11  as  half  the  angle  between  the  two  geodesic 
lines  drawn  through  a  fixed  point  parallel  to  a  given  geodesic  line.  To  deter- 
mine tan  n  he  constructed  the  corresponding  angle  and  lines  on  the  plane.  He 
chose  the  center  of  the  limiting  circle  to  represent  the  fixed  point  and  the  line 
corresponding  to  the  geodesic  bisector  of  the  angle  of  parallelism  for  the  axis  of 
X  so  that  the  coordinates  of  the  extremities  of  the  chord  representing  the  given 
geodesic  line  are  (x,  y)  and  {x,  —  y).     He  could  then  write 


y       Va'-x' 
tan  II  =     = 


and  from  the  equation 


HISTORICAL    DEVELOPMENT    OF    PSEUDOSPHERICAL   SURFACES.  37 


r  =  \^u^  -\-  v^-=a  tanh  -^       (p  =  length  of  geodesic  bisector), 


he  could  obtain  for  x  along  the  axis,  y  =  0,  the  value 

o 
x  =  a  tanh  ^51 
Ji 

so  that  on  the  surface 

1 
tan  n  = 


sinh  ^ 


a  form  equivalent  to  the  one  given  by  Lobatchewsky. 

By  means  of  the  above  equation  he  was  able  to  express  Minding's  equation 
for  the  angle  of  a  geodesic  triangle  in  terms  of  the  angles  of  parallelism  of  the 
sides  and  thus  obtain  the  fundamental  equation  of  the  non-Euclidian  trigonometry 

sin  n  (6)  sin  n  (c)       , 

cos  A  cos  n  (b)  cos  n  (c    + Mt-/    n  -^ 

^    ^  ^    ^  sin  n  (a) 

where  a,  b,  c  are  the  sides  and  A,  B,  C  the  angles  of  the  triangle. 

Finally  he  found  the  area  of  a  triangle  to  be  proportional  to  its  spherical 
deficiency,  a  fact  which  results  from  Gauss"  theorem  that  its  total  curvature  is 
equal  to  the  sum  of  its  angles  minus  tt. 

From  the  theorems  of  pure  geometry  Beltrami  returned  to  the  subject  of  the 
three  different  forms  of  the  pseudospherical  surface  of  rotation.  He  first  found 
the  equation  for  a  geodesic  circle,  or  as  he  called  it  a  geodesic  circumference, 
whose  center  is  the  point  u,  v,  and  whose  radius  is  p,  to  be 

V(a^  —  II'  —  v^)  (a-  —  ?/j  —  ■«,";)  -"' 

and  by  means  of  it  he  deduced  that  the  expression  for  the  linear  element  assumes 
one  of  the  three  different  forms  given  by  Dini, 

ds'  =  ((hr  +  sinh"  -^  dv'), 

ds'  =  {dir  +  e-^'i^dv'), 

ds'  =  (du'^  +  cosh^  p  Jr), 

according  as  to  whether  the  centers  of  the  geodesic  circumferences  chosen  for 
one  family  of  parametric  curves  are  real,  at  infinity  or  ideal,  that  is  whether  on 
the  plane,  the  corresponding  points  lie  within  the  limiting  circle,  on  its  perimeter 
or  entirely  without  it. 

He  also  remarked  upon  the  peculiar  properties  of  the  three  types  of  geodesic 


453:173 


38  E.    M.    CODDINGTON. 

circumferences,  how,  those  of  the  third  tj'pe  with  a  common  center  are  parallel 
to  a  geodesic  curve,  a  property  which  belongs  to  all  geodesic  circles  on  a  sphere, 
but  which  belongs  only  to  geodesic  circles  with  an  ideal  center  on  a  pseudospheri- 
cal  surface ;  how  a  geodesic  circle  of  the  second  type  is  identical  with  what  is 
known  as  the  limiting  circle  or  horicycle  of  Lobatchewsky,  that  is,  a  curved 
line  such  that  all  the  perpendiculars  erected  at  the  middle  point  of  its  chords  are 
parallel  to  each  other ;  how  the  geodesic  lines  orthogonal  to  a  family  of  geodesic 
circles  of  the  first  type  go  through  a  common  point  usually  chosen  for  the  origin 
{n  =  v=0). 

In  this  same  paper,  in  speaking  of  the  three  types  of  surfaces  of  rotation 
which  correspond  to  the  above  three  forms  of  the  linear  element,  he  remarked 
that  in  the  actual  application  of  a  surface  of  rotation  of  the  first  type  upon  a 
pseudospherical  surface  of  a  different  form,  it  is  necessary  to  make  a  slit  in  the 
surface  from  the  point  of  intersection  ( ?<  =  v  =  0 )  of  the  meridian  curves  in 
order  to  apply  the  " pseudospherical  cap "  about  the  point  (^u  =  v  =  0)  upon 
the  second  surface.  He  went  on  to  observe  that  surfaces  of  rotation  of  the  sec- 
ond or  third  type  have  each  a  minimum  parallel  circle,  that  for  the  last  named 
surface,  this  minimum  circle  is  the  geodesic  curve  to  which  all  the  other  par- 
allel circles  are  parallel  and  that  at  equal  distances  from  it,  on  either  side,  lie  two 
maximum  parallel  circles  between  which  lies  the  real  part  of  the  surface  and  that 
when  a  pseudospherical  surface  is  applied  upon  a  surface  of  rotation  of  the  sec- 
ond or  third  type  it  may  be  wrapped  about  it  an  infinite  number  of  times.  These 
properties  though  evident  from  the  drawings  of  pseudospherical  surfaces  at  the 
end  of  Mindiugs  memoir  in  volume  XIX  of  Crelles  Journal,  were  not  de- 
scribed by  him  nor  were  they  spoken  of  in  any  of  the  papers  previously  mentioned. 

8.  Beltrami^'  wrote  a  paper  in  1872  devoted  exclusively  to  the  subject  of  the 
pseudosphere,  making  therein  a  particular  study  of  its  asymptotic  and  geodesic 
curves.  His  theorems  on  geodesic  lines,  following  in  natural  order  after  his 
remarks  in  regard  to  these  curves  in  his  "  Essay  on  the  Interpretation  of  the 
Non-Euclidian  Geometry,"  will  now  be  considered. 

If  the  expression  for  the  linear  element  of  the  pseudosphere  of  curvature 
—  1  /}•■  be  written  in  the  usual  form 

ch-  =  (la-  +  e-"""!'- d4>^ 

where  the  parameter  <7  represents  the  arc  length  of  any  meridian  curve  and  the 
parameter  (^  denotes  the  angle  that  that  meridian  makes  with  a  fixed  meridian 
measured  on  the  plane  of  the  maximum  parallel,  the  general  equation  for  the 
radius  of  geodesic  curvature  of  a  curve  becomes 


HISTORICAL   DEVELOPMENT    OF   PSEUDOSPHERICAL   SURFACES.  39 

Beltrami  obtained  the  equation  for  a  parallel  circle. 

by  putting  p  =  a  constant  in  this  equation  and  then  integrating.  The  denomi- 
nator set  equal  to  zero  gave  him  the  differential  equation  for  a  geodesic  curve 
whose  integral  is 

e'"''  +  ((j)  —  Ij)-  =  (a  +  b-)  .  (n,  6,  c,  f7  are  constants. ) 

If  the  equation  for  a  parallel  circle  be  differentiated  twice  with  respect  to  <j>  it 
will  become 

d'{e"''i'+cj>')_       hd-e^"!' 
d^^  ~       a    d<f>^ 

and  the  combination  of  this  equation  with  the  equation  for  the  radius  of  geo- 
desic curvature  makes  the  latter  assume  the  form 


[-(**)7 


ar  d^e'^i'' 


The  comparison  of  this  equation  with  the  ordinary  one  of  Differential  Calculus 
for  the  radius  of  curvature  of  a  plane  curve  showed  him  that  the  two  become 
identical  ii  (f)  =  x  and  e"'''  =  y.  He  saw,  therefore,  that  the  geodesic  circles  of 
the  surface  may  be  transformed  into  circles  on  a  plane  whose  equation  is 

y-  +  e^  —  ad      h-p- 
cr  a"r- ' 


(^+iy+(--+«y^ 


and  that  the  condition  that  these  circles  should  have  a  center  that  is  a  real  finite 
point,  a  point  at  infinity  or  an  imaginary  point  may  be  expressed  analytically  by 

P  =  r. 

This  projection,  which  is  similar  to  the  stereographic  projection  of  a  sphere, 
has  since  become  very  useful  in  the  investigation  of  pseudospherical  surfaces. 

Beltrami's  first  method  of  projection  or  geodesic  representation  of  a  pseudo- 
spherical  surface  converts  the  geodesic  lines  of  a  surface  into  straight  lines  on 
the  plane ;  his  second  method  transforms  the  geodesic  lines  on  the  surface  into 
circles  on  the  plane. 

Beltrami's  second  method  of  projection  is  conformal,  and  Busse"^  has  shown 
in  a  recent  doctor's  dissertation  that  it  is  only  surfaces  of  constant  curvature 
that  cau  be  conformally  projected  upon  a  plane  in  such  a  way  that  their  geodesic 
curves  become  right  lines  or  the  arcs  of  cii-cles. 

A  very  interesting  deduction  was  made  by  Cayley*"  in  1884  from  the  theo- 
rems contained  in  Beltrami's  two  papers  on  pseudospherical  geometry,  namely. 


40  E.    M.    CODDINGTON. 

that  "  the  Lobatchewskian  geometry  is  a  geometry  such  as  that  of  the  imaginary 
spherical  surface  ^Y-  +  Y-  +  Z"=  —  1  (spoken  of  by  Dini,  page  18)  and  that 
the  imaginary  surface  may  be  bent  without  extension  or  contraction  into  the  real 
surface  considered  by  Beltrami." 

He  remarked  that  this  bending  is  an  "  imaginary  process  "  for  the  points  and 
lines  on  the  first  surface  are  imaginary  and  those  on  the  second  are  real,  while 
the  angles  and  distances  are  real  on  both  surfaces.  He  denoted  the  coordinates 
of  a  point  on  the  imaginary  sphere  of  curvature  —  1  by  Jl,  Y,  Z  ,  and  the 
coordinates  of  a  point  on  a  pseudosphere  of  the  same  curvature  by  x,  y,  z.  He 
was  then  able  to  transfer  the  linear  element  on  the  surface  denoted  by 

ds-  =  dX'+  dY-  +  dZ' 

into  the  linear  elements  of  the  pseudosphere  represented  by 

ds"  =  dx"  +  dy-  +  dz", 

by  means  of  the  three  sets  of  equations 

X^     ..    -\        .,  Y= 


Vl  —  M^  —  U^  '  1     1 


w 


■      A  J-  — M 


l/l-u'-v'  '         ^      1-m' 

X  =  cos  (j)  sin  6 ,         y  =  sin  ^  sin  Q ,         3  =  log  ctn  ^  —  cos  6 , 

where  u  and  v  are  Beltrami's  parameters  which,  when  linearly  connected,  repre- 
sent a  geodesic  curve. 

Having  established  the  fact  that  the  imaginary  sphere  is  transformable  into  a 
real  pseudosphere,  Cayley  proceeded  to  consider  the  geodesic  curves  on  the  first 
surface.  The  equation  of  a  geodesic  curve  on  the  imaginary  sphere  may  be 
written  in  the  form  similar  to  that  of  a  geodesic  curve  on  a  real  sphere 

«A'+  bY -\-  cZ  =^  0  (o,  6,  c,  =  constants) , 

but  Cayley  observed  that  "  since  for  a  point  corresponding  to  a  real  point  of  a 
pseudosphere  ^  is  a  pure  imaginary,  and  Y  and  Z  are  real,  we  see  that  for  a 
geodesic  corresponding  to  a  real  geodesic  of  the  pseudosphere  X  must  be  a 
pure  imaginary  and  X  and  Z  real.  In  order  to  have  aU  the  coefficients  real  he 
therefoi'e  made  the  substitutions 

P  =  iX  -  Y,         Q=  iX  +  Y, 

by  means  of  which  the  equation  becomes 

(  _  iia  -  ii)P  +  (  _  J-,-a  +  ih)Q  +  cZ=0, 
or 

AP  +  BQ+  CZ=0, 


HISTORICAL    DEVELOPMENT   OF   PSEUDOSPHEKICAL    SURFACES.  41 

where  A ,  B  and  C  are  all  real.  Applying  the  same  equations  of  transforma- 
tion to  this  equation  as  he  had  formerly  applied  to  the  imaginary  sphere  to 
deform  it  into  a  pseudosphere  he  found  that  the  equation  assumes  the  form 

A  +  B{f"-^  +  4>')  +(?</,  =  0  , 

which  is  the  form  obtained  by  Beltrami  for  the  geodesic  curves  on  a  pseudo- 
sphere. 

Thus  Cayley  proved  that  the  imaginary  surface  and  the  real  surface  are  so 
related  to  each  other  that  to  every  point  and  to  every  geodesic  line  of  the  one 
there  corresponds  a  point  and  a  geodesic  line  of  the  other. 

Cayley  applied  the  same  method  of  projection  on  the  plane  of  the  greatest 
parallel  to  the  case  of  a  geodesic  line  that  cuts  a  meridian  curve  at  right  angles, 
as  Beltrami  had  applied  to  asymptotic  curves.  By  tracing  the  course  of  the 
projected  line  he  saw  that  it  continues  to  cut  at  right  angles  the  radius  of  the 
maximum  circle  into  which  the  meridian  is  projected,  that  in  the  neighborhood 
of  the  circumference  of  the  circle  it  is  almost  a  straight  line  and  that  the  further 
away  the  point  of  intersection  of  the  meridian  curve  with  the  geodesic  line  on 
the  surface  lies  from  the  plane  of  the  unit  circle,  the  nearer  the  projection  of  the 
line  approaches  the  center  of  the  circle  and  the  more  curved  it  becomes,  while 
the  circle  itself  is  an  envelope  of  geodesic  lines. 

The  question  as  to  whether  Beltrami's  geodesic  projection  of  a  pseudospher- 
ical  surface  on  a  plane  may  represent  the  whole  plane  of  Lobatchewsky's  geom- 
etry was  asked  by  HUbert'*-  and  answered  by  him  in  the  negative,  for  he 
proved  that  it  is  impossible  to  construct  an  analytic  surface  with  constant  nega- 
tive curvature  that  contains  no  singularities.  First  he  assumed  that  such  a  sur- 
face can  be  constructed,  and  showed  that  in  that  case  it  will  be  completely 
covered  by  a  net-work  composed  of  two  families  of  asymptotic  lines,  for  he 
proved  that  no  one  of  these  lines  ever  intersects  one  of  the  curves  of  the  family 
to  which  it  does  not  belong  more  than  once  and  never  intersects  a  member  of 
its  own  family,  and  that  they  have  no  double  points  or  singidarities  of  auj'  kind. 
He  then  saw  that  the  surface  can  be  regarded  as  bounded  by  four  of  these 
asymptotic  lines  no  matter  what  its  extent  may  be  and  that  by  Dini's  theorem 
its  area  will  never  be  greater  than  27r.  On  the  other  hand  he  recalled  Gauss" 
expression  for  the  area  of  a  geodesic  circle  with  radius  /a  on  a  surface  of  curva- 
ture —  l/W, 

and  saw  that,  if  he  supposed  the  surface  to  be  bounded  by  such  a  circle  with  a 
radius  indefinitely  great,  its  area  must  be  greater  than  27r.  Such  an  inconsist- 
ency between  the  two  methods  of  measurement  showed  him  that  there  must  be 
singularities  somewhei-e  on  the  surface,  and  that  therefore  the  projection  of  such 
an  analytic  surface  does  not  repi'esent  the  whole  of  Lobatschewsky's  plane. 


42  E.    M.    CODDINGTON. 

9.  It  remained  for  Klein  "^  to  reconcile  these  two  geometries,  the  Pseudo- 
spherical  geometry  of  Beltrami  and  the  non-Euclidian  geometry  of  Lobatchewsky 
with  still  a  third,  the  Metrical  geometry  of  Cayley. 

Cayley"  first  originated  this  geometry  in  1859,  as  a  result  of  his  studies  on 
the  projective  properties  of  points,  lines  and  planes.  In  this  connection,  he 
considered  the  distance  between  two  points  (x,,  ^/,)  and  (x.,,  y^)  on  a  plane  as 
denoted  by  the  formula 

cos  T    /   ,       =; 

Vx\  +  y\  Vxl  +  yl 

and  between  two  points  {x^,  y^,  ^^),  (x^,  y^,  s,)  ^^  ^  sphere  by  the  formula 

x^x^  +  y^y^  +  z^Z2 

yxi  +  7/-  +  z-  V  x:+yi+  8- 

where  x^,y^,z^,  x^ ,y^,  s, ,  are  ordinary  rectilinear  coordinates.  Cayley  observed 
that  the  first  formula  might  represent  the  angle  between  the  polars  of  the  points 
with  respect  to  a  conic  whose  equation  is 

-.r  +  ?/-  =  0 

and  the  second,  the  angle  between  the  polars  of  the  points  with  respect  to  a 
spherical  conic  whose  equation  is 

x^  +  f  +  z-=  0. 

Therefore  in  order  to  measure  the  distance  between  any  two  points  on  a  plane, 

he  assumed  an  imaginary  conic  which  he  called  the  "absolute"  and  formed  the 

expression 

(a,  6,  c(j.t;i,  3/i,  sJ.T,,  i/,,  x^) 

cos~  — ^ —  —  — 

l/(a,  b,  cjjxp  y,,  3,)-i'(a,  b,  c\x^,y^,zj' 

where  x^ ,  y^,  z^  and  .^2 ,  y, '  ^2  ^'"®  *'^®  homogeneous  coordinates  of  the  points  and 

(a,  6,  cjj.x,  y,  «)'"  =  ax'  -\-  by-  +  cs'  =  0, 

is  the  equation  of  the  conic. 

He  observed  the  fact  that  the  two  points  together  with  the  points  of  inter- 
section of  their  binding  line  with  the  absolute  are  in  involution  and  that  two 
systems  in  involution  are  homograpically  related.  He  also  discovered  that  if 
line  coordinates  are  used  instead  of  point  coordinates  exactly  the  same  formula 
wiU  measui-e  the  angle  between  two  lines,  and  that,  in  that  case,  these  lines  and 
the  tangents  drawn  to  the  absolute  from  their  point  of  intersection  are  in  invo- 
lution . 

Cayley  himself  never  applied  his  theories  to  the  case  of  pseudospherical  sur- 
faces but  Klein,  perceiving  that  if  he  assumed  a  general  formula. 


HISTORICAL    DEVELOPMENT   OF   TSEUDOSPHERICAL    SURFACES.  43 

l/(a,  b,  cjjxj,  y^,  s,)V(a,  6,  c^a;^,  y^-  ^if' 

for  the  measure  of  distance  between  two  points  on  a  plane,  he  could  derive  from 
it  Cay  ley's  expression  by  putting  for  C  the  value  —  i/2  and  the  expression  of  the 
Euclidian  geometry  by  letting  C  become  infinitely  great  introduced  as  a  third 
value  for  C  a  real  finite  quantity,  and  thus  obtained  an  expression  that  satisfies 
the  requirements  of  the  theorems  of  the  non-Euclidian  geometry, 
Klein  denoted  the  absolute  in  homogeneous  point  coordinates  by 

n  =  o, 

so  that  the  general  expression  for  the  distance  between  two  points  re  and  y 
becomes 

n 

2iCcos-i— ^^=, 

where  fi  and  O  are  the  expressions  which  result  when  the  coordinates 
Xj ,  x^,  «.j ,  of  the  point  x  or  the  coordinates  y^ ,  y^ ,  y^  of  the  point  y  are  set  in 
fl ,  and  n^^  is  the  consequence  of  putting  the  coordinates  of  x  in  the  polar  of  y 
or  conversely. 

He  changed  this  general  expression  into  the  equivalent  form 


n   +  i/a=  -  ft  n 

Clog   '"     \^' "~'\ 

"ft   -  V^-  -  ft  ft 

'V  Jfy  I"     yy 

and  observed  that  the  expression  under  the  sign  of  the  logarithm  is  the  anhar- 
monic  ratio  formed  by  the  two  points  x  and  y  and  the  points  of  intersection 
with  the  absolute  of  the  line  joining  them. 

He  obtained  a  similar  expression  for  the  distance  between  two  lines  repre- 
sented by  u  and  u,  namely 

C"  lo?     ""  ^         ""        ""    ■"" 


when  <t>  =  0  is  the  equation  of  the  absolute  in  homogeneous  line  coordinates. 
He  saw  that  both  expressions  under  the  sigu  of  the  logarithm  are  anharmonic 
ratios,  the  first  formed  of  four  points,  the  second  of  four  lines,  each  of  which, 
according  to  Cayley,  belongs  to  a  system  in  involution,  and  that  therefore  every 
point  in  a  line  except  the  points  of  its  intersection  with  the  absolute  may  be 
linearlj'  transformed  in  every  other  point  and  every  ray  in  a  pencil,  except  the 
two  tangent  to  the  absolute,  may  be  linearly  transformed  into  every  other  ray. 

Klein  investigated  the  nature  of  the  absolute  and  discovered  its  characteristic 
properties;  first,  that  since  for  an  imaginary  value  of    C  it  is  imaginary,  for  a 


44  E.    M.    CODDINGTON. 

real  value  of  C  it  must  be  real,  and  that,  in  that  case,  since  only  real  distances 
are  considered,  the  auharmonic  ratio  is  positive  and  all  real  points  lie  within  its 
circumference ;  second,  that  it  lies  at  infinity,  for  if  a  conic  is  assumed  to  be  a 
circle  with  x  as  its  center  and  y  a  point  on  its  circumference,  its  radius  will  be 
by  Cayley's  formula  equal  to 

2/ecos-J^, 

JX        yy 

and  will  become  infinitely  great  when  y  lies  on  the  circumference  of  the  conic, 
0^=0;  third,  that  it  is  impossible  to  determine  the  region  outside  of  the  abso- 
lute, "  the  ideal  region,"  for  by  means  of  a  linear  transformation  a  man  starting 
from  any  point  within  the  conic  to  walk  to  its  infinitely  far-off  circumference  at 
a  uniform  velocity  will  never  reach  it,  much  less  then  will  he  know  what  lies 
beyond. 

He  therefore  concerned  himself  only  with  the  points  and  angles  within  the 
absolute  and  saw  that  for  every  line  the  fundamental  elements  are  real,  but  that 
for  each  pencil  of  rays  they  are  imaginary,  since  no  real  tangent  can  be  drawn 
from  an  interior  point  to  the  conic.  He  then  put  for  C  the  value  i'/2,  so  that 
the  sum  of  the  angles  about  a  point  is  the  same  as  in  ordinary  plane  geometr}'. 

This  description  of  the  absolute,  that  it  is  a  real  circle  at  infinity  within  whose 
circumference  lie  all  real  points,  is  exactlj'  the  same  as  the  definition  of  Belt- 
rami's limiting  circle,  and  Beltrami's  expression 


R  ^      a  +  Vu"-  +  v" 
-^  f  —  1  tr  +  v^ 

for  the  length  of  a  geodesic  line  from  the  center  of  this  circle  is  exactly  the  same 
as  Klein's,  if  C  =  Ji/2.  Consequently  the  propositions  proved  by  Beltrami  with 
respect  to  parallel  lines,  the  angle  of  parallelism  and  trigonometrical  ratios  belong 
equally  to  the  metrical  geometry  and  may  be  solved  by  means  of  figures  drawn 
in  the  plane  of  the  absolute. 

This  geometry  Klein  called  the  Hyperbolic  geometry  and  the  spherical  and 
Euclidean  geometries  he  called  Elliptic  and  Parabolic  respectively,  making  the 
distinction  between  them  depend  upon  whether  the  right  line  has  two  real, 
imaginary,  or  coincident  points  at  infinity.  He  called  the  measure  of  distance 
of  the  Hyperbolic  and  Elliptic  geometries,  the  general  metrical  determination, 
and  that  of  the  Parabolic,  the  special  metrical  determination.  He  remarked 
that  the  two  may  coincide  at  a  point  or  in  the  neighborhood  of  a  point,  but  that 
at  points  at  a  distance  from  the  point  of  contact,  the  general  metrical  determina- 
tion is  greater  or  less  than  the  special,  according  as  to  whether  the  fundamental 
conic  is  imaginary  or  real.  He  designated  as  measure  of  curvature  the  gi-eatness 
of  the  respective  gain  and  loss,  and  found  that  it  is  the  same  at  every  point,  and 
that  it  is  equal  to  —  1/4  C'. 


HISTORICAL   DEVELOPMENT   OF   PSEUDOSPHERICAL    SURFACES.  45 

He  regarded  all  the  points  and  lines  on  the  plane  as  the  projections  of  lines 
and  planes  iu  space  and  the  absolute  as  the  section  of  a  cone  whose  vertex  lies 
at  a  determined  point  in  space  and  which  passes  through  the  circle  of  infinity, 
and  was  able  to  prove  that  projective  geometry  can  be  completely  developed, 
although  absolutely  free  from  the  question  of  metrical  determination.  He  thus 
showed  that  the  hyperbolic  geometry,  since  it  has  a  real  value  for  C  is  the  geom- 
etry of  surfaces  with  constant  negative  curvatui-e  and  that  the  non-Enclidian 
geometry,  the  pseudospherical  geometry  and  the  hyperbolic  geometry  are  essen- 
tially one  and  the  same. 


II 

THE  SURFACE  OF  CENTERS  AND  THE  TRANSFORMATION 
OF  PSEUDOSPHERICAL  SURFACES. 


1.  The  theorem  that  oo'  new  pseudospherical  surfaces  may  be  derived  from 
oue  that  is  known  and  the  geometrical  method  for  the  determination  of  the  new 
surfaces  were  derived  by  Biauchi**'  for  a  simple  case  only  in  1879.  In  1881  the 
theorem  was  developed  analytically  so  as  to  apply  to  a  more  general  case  by 
Backlund*"  and  was  geometrically  interpreted  for  this  general  case  by  Bianchi"" 
in  1887. 

In  its  generalized  form  the  theorem  may  be  stated  as  follows :  if  on  a  surface 
of  constant  negative  curvature  —  1/ H"  a  system  of  lines  be  chosen  whose  prin 
cipal  normals  at  every  point  make  a  constant  angle  (jr/2  —  <t)  with  the  normal 
to  the  surface  at  that  point,  and  if  tangent  lines  be  drawn  to  these  curves  on  each 
of  which  a  constant  length  i?  cos  rr  is  measured  off,  the  extreme  points  of  the 
constant  lengths  on  these  tangent  lines  will  lie  upon  a  second  surface  which  has 
also  constant  negative  curvature  —  1/^-  and  which  can  be  completely  determined 
when  the  first  surface  and  <7  are  known. 

This  theorem  as  here  written  was  not  announced  all  at  one  time  nor  was  it 
the  work  of  a  single  man,  but  it  is  the  result  of  the  discoveries  of  many  other 
theorems  and  it  represents  the  labor  of  many  men  of  different  nationalities. 

It  is  necessary  in  tracing  the  gradual  development  of  the  theory  which  lies 
beneath  it,  to  go  back  to  Kummer's"'  treatise  on  ray  systems,  for  Bianchi''-' 
observed,  that  the  tangent  lines  may  be  regarded  as  forming  a  congruence  of 
right  lines  for  which  the  two  pseudospherical  surfaces  are  the  focal  surfaces. 
Thus  he  pointed  out  that  the  whole  theory  of  deriving  new  pseudospherical  sur- 
faces from  a  given  one  is  dependent  upon  Kummer's  theorems. 

2.  Kummer's  paper  entitled  The  General  Theory  of  Congruences  of  Right 
Lines  was  published  in  1860.  His  definitions  and  theorems  which  were  afterward 
used  by  Bianchi  in  the  development  of  this  theory  may  be  briefly  stated  as  fol- 
lows :  suppose  a  system  of  rays  to  be  comj)osed  of  oo-  right  lines,  if  all  the  rays 
of  this  system  pass  through  a  surface  arbitrarily  chosen  as  a  surface  of  reference, 
each  ray  is  determined  by  its  direction  cosines  X,  Y,  Z  and  the  coordinates 
a;,  y,  z  of  its  point  of  intersection  with  the  surface  with  reference  to  a  set  of 
rectangular  coordinates  in  space.     If  the  linear  element  of  the  surface  of  refer- 

46 


HISTORICAL    DEVELOPMENT   OF    PSEUDOSPHERICAL    SURFACES.  47 

ence  be  referred  to  curvilinear  coordinates  v  =  a  constant  and  v  =  a.  constant, 
of  parameters,  the  quantities  x,  y,  z,  A',  J^,  Z  may  be  expressed  as  functions 
of  II  and  I'. 

The  abscissa  of  a  point  in  space  is  its  distance  from  the  initial  surface  meas- 
ured along  the  ray  on  which  it  lies. 

Upon  every  ray  of  the  system  there  lie  five  points  which  are  of  especial  im- 
portance—  the  two  limiting  points,  the  two  focal  points  and  the  middle  point. 
The  limiting  points  may  be  defined  as  follows :  if  to  any  ray  («,  v)  the  lines  of 
its  shortest  distance  from  all  the  rays  of  the  system  that  are  infinitely  near  it  be 
drawn,  the  foot  points  on  the  ray  of  these  lines  of  shortest  distance  will  have, 
the  one  a  maximum,  the  other  a  minimum  abscissa,  the  foot  points  of  all  the 
other  lines  of  shortest  distance  to  the  ray  will  lie  between  these  two  and  they 
are  called  the  limiting  points.  The  lines  of  shortest  distance  drawn  to  the  lim- 
iting points  of  a  ray  from  the  two  rays  infinitely  near  it  are  at  right  angles  to 
each  other,  consequently  the  two  planes  drawn  through  the  ray  normal  to  these 
two  lines  of  shortest  distance  respectively  are  perpendicular  to  each  other. 

The  surface  which  is  the  locus  of  the  limiting  points  with  the  maximum 
abscissaj  and  the  surface  which  is  the  locus  of  the  limiting  points  with  the  mini- 
mum abscissa  are  called  the  principal  surfaces. 

The  point  on  the  raj-  which  is  half  way  between  its  limiting  ])oints  is  called 
its  middle  point. 

The  rays  of  the  system  themselves  form  two  families  of  developable  surfaces, 
real  or  imaginary,  and  two  surfaces  pass  throught  every  ray.  The  curves  of 
intersection  of  the  two  families  of  developable  surfaces  with  the  surface  of 
reference  are  usually  chosen  for  the  parametric  lines  u  and  v  respectively. 

The  surface  which  is  the  locus  of  the  edges  of  i-egression  of  the  developable 
surfaces  of  either  of  these  families  is  called  a  focal  surface.  Every  ray  is  tan- 
gent to  both  focal  surfaces  and  its  two  points  of  contact  with  the  two  focal  sur- 
faces are  called  its  focal  points. 

The  curves  along  which  the  developable  surfaces  of  the  one  family  meet  a 
focal  surface  are  conjugate  to  the  curves  along  which  the  developable  surfaces  of 
the  other  family  meet  that  same  focal  surface. 

In  particular,  Kummer  proved  that  the  condition  that  the  right  lines  of  a 
system  shall  become  the  normals  to  a  surface,  or  rather  to  an  infinite  number  of 
parallel  surfaces,  is  that  the  focal  surfaces  shall  coincide  with  the  principal  sur- 
faces of  the  ray  sj'stem  and  thus  form  the  two  nappes  of  the  surface  of  centers 
or  evolute  surface  of  the  parallel  surfaces.  A  special  case  of  the  transformation 
of  pseudospherical  surfaces  occurs  when  the  tangents  to  the  given  surface  become 
the  normals  to  a  series  of  parallel  surfaces,  but  before  taking  up  this  subject  it 
is  necessary  to  make  a  study  of  the  well  known  theorems  which  were  derived  by 
Weingarten,  Beltrami  and  Dini  with  respect  to  the  involute  and  evolute  surfaces 
of  surfaces  with  constant  negative  curvature. 


48  E.    M.    CODDINGTON. 

3.  Weingarten  investigated  a  class  of  surfaces  which  are  distinguished  by  the 
property  that  at  every  point  one  of  their  principal  radii  of  curvature  is  a  func- 
tion of  the  other,  and  which  are  now  called  TF-surfaces.  Surfaces  with  con- 
stant negative  curvature  come  under  this  heading  and  they  are  also  connected 
with  this  class  of  surfaces  by  the  fact  that  every  pseudospherical  surface  is  a 
nappe  of  the  evolute  of  a  TF-surface. 

Weingarten  wrote  two  papers  on  these  surfaces  which  ajipeared  in  Crelle's 
Journal.  In  the  first,"  published  in  1861,  he  proved  the  theorem  that  the  two 
nappes  of  the  evolute  surface  of  a  surface  whose  principal  radii  of  curvature  R^ 
and  R^  are  bound  together  by  a  relation,  i?,  =  ^(i?; ),  are  each  applicable  upon  a 
surface  of  rotation,  and  that,  if  the  first  nappe  corresponds  to  the  lines  of  curva- 
ture ?<  =  a  constant  of  the  involute  surface  along  which  the  principal  radius  of 
curvature  is  denoted  by  R^  and  the  second  nappe  corresponds  to  the  lines  of 
curvature  w  =  a  constant  of  the  involute  surface  along  with  the  principal  radius 
of  curvature  is  represented  by  i?,,  the  expression  for  the  linear  elements  of  the 
first  nappe  assumes  the  form 

dsi  =  (IR;  +  c'-^  ^^-^'-du- , 
and  that  of  the  second  nappe  the  form 

dsl  =  dRl  +  c-^  ^-^-^^dvK 

He  also  demonstrated  the  converse  of  this  theorem,  that,  if  a  surface  is  develop- 
able upon  a  surface  of  rotation,  it  may  be  considered  as  a  nappe  of  the  evolute 
of  a  surface  whose  radii  of  curvature  are  functionally  related. 

As  a  special  illustration  of  the  converse  of  this  theorem,  Weingarten  sup- 
posed one  of  the  nappes  of  surface  of  centers  to  be  developable  upon  a  cate- 
noid,  and  proved  that  the  relation  which  binds  together  the  principal  radii  of 
curvature  of  its  involute  surface  is 

R^R^=  — u^  (a  =  a  constant), 

thus  proving  in  an  inverse  fashion  that  a  nappe  of  the  evolute  surface  of  a 
pseudosj)herical  surface  is  ajjplicable  upon  a  cateuoid. 

In  his  second ""  paper  on  TF-surfaces,  published  in  1863,  he  demonstrated  a 
second  theorem  —  that  the  spherical  i-epresentation  of  the  linear  element  of  such 
a  surface  referred  to  its  lines  of  curvature  for  parameters  may  be  denoted  by 


ds'  =  -Y^.,  die'  -f-  I    , ,-  „,  I  dv' 
A-  \<I>{A)J 


where  cj)  and  -ff'are  functions  of  w  and  v  of  such  a  nature  that  they  define  the 
principal  radii  of  curvature  R^ ,  i?.,  of  the  surface  by  means  of  the  equations, 

^j  =  ^(A"),         R.,  =  4>{Ii)-K(ji'{K). 


HISTORICAL    DEVELOPMENT   OF    PSEUDOSPHERIOAL   SURFACES.  49 

The  expression  for  the  linear  element  of  the  Tf-surface  itself,  when  exjjressed  in 
this  notation,  then  becomes 

*..  (iim,,„-+(i'('^-\7/sf^''^)'d^. 


P) 


K    )-   ^\  ^\K) 

Beltrami  -"  in  a  series  of  articles  on  the  application  of  analysis  to  geometry, 
published  in  1865,  proved  both  Weingarten's  theorem  and  its  converse  and  for 
the  latter  found  that  a  ruled  helicoidal  surface  forms  a  case  of  exception,  for 
although  this  surface  is  applicable  upon  a  surface  of  rotation,  it  cannot  be  the 
nappe  of  a  surface  of  centers.  He  showed  at  this  time  that  the  curves  on  the 
nappe  of  a  surface  of  centers  which  are  enveloped  by  the  normals  to  the  involute 
surface  are  geodesic  lines,  he  therefore  remarked  that  "if  the  geodesic  lines  of  an 
evolute  surface  become  right  lines  the  tangents  to  them  at  every  point  instead 
of  filling  all  space  reduce  to  a  system  of  straight  lines  with  a  single  parameter 
and  are  not  sufficient  to  generate  an  orthogonal  surface  " ;  he  discovered  rather 
that,  then,  the  geodesies  themselves  can  generate  a  ruled  surface  which  if  it  is 
applicable  upon  a  surface  of  rotation  is  applicable  upon  the  minimal  surface  of 
rotation,  the  cateuoid,  and  is  parallel  to  a  series  of  pseudospherical  surfaces 
instead  of  being  a  nappe  of  their  evolute  surface. 

Diui-'  also  investigated  this  case  of  exception  to  the  converse  of  Weingarten's 
theorem  in  his  paper  on  helicoidal  surfaces,  in  the  same  year.  He  found  that 
the  ruled  helicoid  that  is  applicable  upon  a  eatenoid  is  a  screw-surface  generated 
by  a  right  line  that  moves  along  a  helix  lying  on  a  cylinder,  making  a  right 
angle  with  the  helix  at  every  point  and  a  constant  angle  with  the  cylinder,  and 
that  it  may  be  regarded  as  the  locus  of  the  normals  of  another  helicoidal  surface 
upon  which  these  same  helices  lie. 

In  the  same  treatise  on  the  application  of  Analysis  to  Geometry-"  Beltrami 
made  known  several  very  important  theorems  concerning  the  surface  of  centers. 
He  denoted  a  surface  whose  principal  radii  of  curvature  are  functionally  related 
by  aS',  its  lines  of  curvature  by  m  =  a  constant  and  y  =  a  constant,  its  principal 
radii  of  curvature  and  the  two  nappes  of  its  evolute  surface  corresponding  to 
those  lines  of  curvature  respectively  by  R^,  R^  and  S^,  S^. 

First,  he  demonstrated  the  general  theorem  that  if  two  systems  of  curves,  one 
of  which  is  composed  of  geodesic  lines,  be  conjugate  to  each  other  and  if  tan- 
gents be  drawn  to  two  of  the  geodesic  lines  that  lie  infinitely  near  each  other  at 
points  a,  and  «.,  where  they  meet  a  curve  of  the  other  system,  these  tangents  will 
meet  at  a  point  which  is  the  center  of  geodesic  curvature  of  a  curve  which  passes 
through  the  point  «, ,  and  is  orthogonal  to  all  the  geodesic  curves  of  the  first 
system.  Applying  this  proposition  to  the  case  of  the  nappes,  ;S'i ,  S^  of  the  evo- 
lute surface  of  a  IF-surface  he  obtained  the  results  first,  that,  since  the  normals 
to  the  surface  *S'  taken  along  its  lines  of  curvature  u  =  a.  constant  touch  the  first 


50  E.    M.    CODDINGTON. 

nappe  of  the  surface  of  centers,  S^ ,  along  a  family  of  geodesic  lines,  which  are  the 
evolutes  of  these  lines  of  curvature  and  which  may  also  be  denoted  by  m  =  a  con- 
stant, and  the  normals  to  the  surface  taken  along  the  lines  of  curvature  v  =  a  con- 
stant, are  tangent  to  this  same  nappe  along  curves  which,  as  Kummer  has  shown,* 
are  conjugate  to  the  geodesic  lines  u  =  a  constant,  S^  is  the  locus  of  the  centers  of 
geodesic  curvature  of  the  orthogonal  trajectories  of  the  geodesic  lines  ?/.  =  a  con- 
stant on  S^  and,  conversely,  the  centers  of  geodesic  curvature  of  the  orthogonal 
trajectories  of  the  geodesic  lines  corresponding  to  u  =  a  constant  on  S^  lie  on 
/Sj ;  second,  that  the  difference  between  the  principal  radii  of  curvature  of  the 
involute  surface  S  at  any  point  J^  is  equal  to  the  radius  of  geodesic  curvature 
at  a  corresponding  point  of  the  orthogonal  trajectory  of  the  curves  on  either 
nappe  which  are  the  evolutes  of  the  lines  of  curvature  of  the  surface  S,  thirdly, 
that,  when  k^  denotes  the  arc  length  of  a  curve  in  the  nappe  S\  which  goes 
through  any  point  ^j  corresponding  to  p  and  which  is  the  evolute  of  a  line  of 
curvature  ?«  =  a  constant  in  the  surface  S  and  when  p  denotes  the  radius  of 
geodesic  curvature  for  the  point  p  of  a  curve  orthogonal  to  Wj  and  going  through 
the  point  p ,  the  principal  radii  of  curvature  7^  and  i?.,  of  the  surface  S  a.t  P 
are  given  by  the  equations 

Beltrami's^''  direct  contribution  to  the  subject  of  pseudospherical  sui-faces  at 
this  time  consisted  In  the  determination  of  their  evolute  and  involute  surfaces. 
He  first  found  the  equation  of  relation  connecting  the  principal  radii  of  curva- 
ture i?j  and  i?2  °^  ^'^y   TF-surface  defined  by  the  equation 

and  the  principal  radii  of  curvature  Ii[ ,  i?^  of  the  surface  of  rotation  on  which  is 
developable  one  of  the  nappes  of  its  evolute  surface.  He  wrote  the  equations  for 
Ii[  and  i?.',  in  the  usual  form  for  the  principal  radii  of  curvature  of  the  surface 
of  rotation. 


aI-(^) 


jX  ,    —    "~~  70  >  Jlij 


1-  fJ2  y  -2-       r 


where  r^  =  the  radius  of  a  parallel  circle  and  ?/,  =  the  arc  of  a  meridian  curve. 
Substituting  in  these  equations  the  expressions  for  i\  and  drjdu^ , 


=  e 


/—""i—  clr,  r,  (Fr,  1     du, 

'  J?/,       ?(,  -  <|)  ( ?<i ) '  du\       (7<i  -  ^  ( ?/, )  )= 


*  P.  47. 

t  Beltrami  uses  contrarj'  signs  for  ii,  and  «,  in  accordance  with  bis  definition  of  geodesic 
curvature. 


HISTORICAL    DEVELOPMENT    OF    PSEUDOSPHERICAL    SURFACES.  51 

derived  from  Weingarten's  form  for  the  linear  element  of  the  surface  of  rota- 
tion, he  obtained  the  two  equations 

To  find  the  evolute  surface  of  a  surface  of  constant  negative  curvature  —  K'^ 
he  put  in  the  equations  for  R[  and  R., . 

and  found  that  they  reduce  to 

the  equation  of  a  minimal  surface,  which  shows  that  the  nappes  of  the  evolute 
surface  of  a  pseudospherical  surface  are  applicable  upon  a  catenoid.  To  find 
the  involute  surface  of  a  surface  of  constant  negative  curvature  —  A'",  he  made 
in  these  same  equations  of  relation  the  substitution  R[  R'^  =  —A'-  and  obtained 
the  equation 

where  A  and  B  are  constants  of  integration. 

When  neither  A  nor  B  is  zero,  he  denoted  their  ratio  by 

±  e""' ,  ( 7«  ^  arbitrary  constant ) , 

so  that 

^,  -  i?,  =  K  tanh  (  m  -  ^-'  ) ' 

R,-  R^  =  K  ctnh  (  "'  —  ;^  ) ' 

according  as  the  upper  or  lower  sign  of  e"'"  is  taken.     When  either  A  or  B  is 

zero,  his  equation  reduces  to 

R^-R^  =  ±  A', 

which  shows  that  the  surface  of  rotation  has  jjarallels  with  constant  geodesic 
curvature  and  must  be  a  pseudosphere.  Beltrami,  therefore,  announced  the 
theorem  "  that  the  evolute  surface  of  surfaces  which  have  at  every  point  the  dif- 
ference of  their  principal  radii  of  curvature  constant  and  equal  to  A"  is  a  sur- 
face of  constant  negative  curvature  —  1/A'-." 

This  same  theorem  was  proved  in  a  more  direct  way  by  Enneper''  in  1868. 
He  used  the  subscripts  ( 1 )  and  ( 2 )  to  denote  the  quantities  on  the  first  and 
second  nappe  respectively  of  an  evolute  surface,  and  from  the  equations  for  the 
coordinates  of  a  point  on  each  obtained  all  the  coefficients  of  their  two  funda- 


52  E.    M.    CODDIXGTON. 

mental  forms.      He  then  found  for  the  measure  of  curvature  of  each,  K^  and  K^ , 
the  expressions 

'  ~       {R,-  R„y  dR^ '         ^^"'-       ( i?,  -  R„  f  dR^  • 

As  a  particular  case  he  put 

i?2  —  R^  =  &  constant, 

which  necessitates  that  the  curvature  of  each  nappe  is  constant  and  negative. 

4.  The  correspondence  between  the  lines  of  curvature  and  that  of  the  asymp- 
totic curves  on  the  two  nappes  of  a  surface  of  a  TF-surface  was  demonstrated  by 
Ribaucour™  in  a  paper  read  before  the  French  Academy  in  1872.  He  wrote 
the  expression  for  the  linear  element  of  the  initial  surface  referred  to  its  lines 
of  curvature  as  coordinates  in  the  form 

ds'  =  fhhr  +  g'dvr, 

and  defined  its  principal  radii  of  curvature  by  the  expressions* 

R   =  ■—  ,  R   ='y  •         {a,  b=^ functions  of  u  and  v.) 


a 


h 


He  obtained  for  the  lines  of  curvature  on  the  first  nappe  the  equation 


dR^  adu 


(R„-R,)bdv~  Idg   ^        15/-       ' 

^     ^  ^^  -r,~dv ~-du 

J  cu  g  civ 

and  for  those  on  the  second  nappe  the  equation 

dR^  hdv 


{R,-R,)adti~  1  dg  ^        1  ^/  7    ■ 
J  cu  g  cv 

Therefore,  the  condition  that  the  lines  on  the  nappes  shall  correspond  is 

dR^  =  dR^ 

or  that  R„  —  ^j  =  a  constant,  and  both  nappes  have  constant  negative  curva- 
ture, he  wrote  the  equations  for  the  asymptotic  lines  in  the  form 

1  dg  dR^^  ^      1  df  BR.  ^  „      ^ 

/  cu      cv  g  f^v     cu 

and 

Idg   BR  Idf  BR 

-TTa^--^—  oi)*±  -5 ^-^dir  —  0, 

J  cu     cu  g  Bv     Bu 

so  that  the  condition  for  their  correspondence  is 

BR^    BR^      BR^    BR. 


Bv       Bti         Bu       Bv 
or  i?j  and  R^  are  functions  of  each  other. 
*  Bianchi,!'"  §  64, 127,  128. 


=  0, 


I 


HISTORICAL    DEVELOPMENT    OF   PSEUDD^PHERICAL    SURFACES.  53 

Ribaucour"'  ^'  had  at  this  time  already  communicated  to  the  Academy  a 
series  of  propositions  in  regard  to  the  class  of  surfaces  which  he  called  "  cyclic," 
and  defined  as  a  system  of  surfaces  that  have  a  family  of  circles  for  orthogonal 
trajectories.  By  so  doing  he  virtually  laid  the  foundation  of  the  Transforma- 
tion Theory  that  is,  the  theory  of  deriving  an  infinite  number  of  surfaces  of 
negative  constant  curvature  from  one  that  is  known,  but  the  connection  l)etween 
this  theory  and  a  cyclic  system  was  not  seen  until  Backlund  *  pointed  it  out  ten 
years  afterwards. 

Ribaucour's  propositions  were,  first,  that  if  the  family  of  circles  are  orthogonal 
to  three  surfaces,  they  will  be  orthogonal  to  an  infinity  of  them ;  that  these 
surfaces  form  part  of  a  triply  orthogonal  system  whose  other  two  families  are 
composed  of  the  envelopes  of  spheres  and  that  they  are  intimately  connected 
with  the  theory  of  deformation  ;  second,  that  in  order  to  find  all  the  trajectory 
surfaces  when  one  {A)  is  determined,  it  is  necessary  to  know  a  function  Z  on 
( ^1 )  which  satisfies  the  partial  differential  equation, 

d'Z  _  JL  ^  ^       1    dlf^    dZ 

where  the  linear  element  of  the  surface  is  denoted  by 

dr  =  11 -dp-  +  H'ldpi , 

and  that  this  equation  is  integrable  at  once  when  the  lines  of  curvature  for  (^A) 
are  geodesic  cii-cles  ;  third,  that,  in  the  special  case,  when  the  circles  lie  in  the 
tangent  plane  of  a  given  surface  and  have  all  the  same  constant  radius,  the 
surfaces  orthogonal  to  these  circles  are  all  applicable  upon  a  surface  {A)  which  is 
itself  applicable  upon  a  pseudosphere ;  fourth,  that  if  a  system  of  curves  are 
normal  to  family  of  surfaces  that  form  part  of  an  orthogonal  system,  the  osculat- 
ing circles  of  those  curves  will  be  normal  to  a  family  of  surfaces  that  belongs 
to  a  cyclic  system. 

5.  Proofs  of  these  theorems  were  worked  out  ten  years  later  by  L.  Bianchi  '■^' '"'' 
89, 90, 93, 94  jjj  ^  series  of  elaborate  treatises,  but  during  these  ten  years  the  transforma- 
tion theory  itself  was  formally  established,  by  means  of  which  all  pseudospherical 
surfaces  may  be  obtained  by  quadrature  alone  when  one  is  known.  Bianchi  '''•^•'''"■* 
gave  the  first  conception  of  the  theory  in  1879  and  published  a  more  concise 
statement  of  his  results  in  1880.  He  regarded  a  pseudospherical  surface  as  one 
of  the  nappes  of  the  evolute  of  an  unknown  Tf-surface  and  proved  that  the 
second  nappe,  which  he  called  the  "complementary  surface,"  may  be  one  of  an 
infinity  of  surfaces,  each  corresponding  to  a  family  of  geodesic  lines  on  the  first 
nappe. 

From  the  theorems  of  Weingarten  and  Beltrami  he  knew  that  the  tangents 

*  Page  59. 


54 


E.    M.    CODDINGTON. 


common  to  the  two  nappes  touch  them  along  geodesic  lines,  that  both  nappes 
are  developable  into  surfaces  of  rotation  and  that  either  is  the  locus  of  the 
centers  of  geodesic  curvature  of  the  orthogonal  trajectories  of  the  geodesic  lines 
on  the  other  to  which  the  normals  of  the  involute  surface  are  tangent.  Choos- 
ing, then,  a  family  of  geodesic  lines  on  the  first  nappe  that  became  meridian 
curves  vi^hen  the  surface  is  deformed  into  one  of  rotation,  he  obtained  the  second 
nappe  by  the  following  rule :  "  On  each  tangent  to  the  geodesies  of  a  system  on 
the  first  nappe  S^  cut  off  a  portion  equal  to  the  radius  of  geodesic  curvature  of 
the  trajectory  orthogonal  to  the  geodesic  at  that  point.  The  locus  of  the  new 
extremes  is  the  surface  S„  complementary  to  S^." 

Since  every  surface  of  constant  negative  curvature  possesses  three  different 
kinds  of  systems  of  geodesic  lines,  those  that  go  out  from  a  real  finite  point, 
those  that  go  out  from  a  point  at  infinity  and  those  that  go  out  from  an  imag- 
inary point,  Bianchi  discovered  that  the  surface  complementary  to  a  pseudo- 
spherical  surface  is  applicable  upon  a  rotation  surface  of  one  of  three  different 
forms  depending  upon  which  kind  of  geodesic  lines  are  selected  on  the  original 
surface. 

He  wrote  down  the  equations  of  the  relation  between  the  radii  of  curvature 
of  the  involute  surface  of  surfaces  of  curvature  —  l/a"  for  the  three  cases,  as 
they  were  given  by  Beltrami,* 


i?,  =  a  tanh 
^,  -  E,  =  a , 
7?   —  li,  =  a  ctnh 


w  +  c 


Substituting  these  values  in  Weingarten's  formula  for  the  linear  element  of 
the  second  nappe, 

he  found  the  three  corresponding  expressions  for  the  linear  element  to  be 

7?  R 

dS;  =  tanh^  — '  dR:  +  sech=  — '  dv' . 

dSl  =  dR\  +  e-'^^"'dv\ 


.R 


R, 


dSl  =  ctuh^  -^  dR\  -f  csch^  -^  dv\ 

In  regard  to  the  profile  curve  he  found  that,  although  the  equation  of  z  is 
always  the  same 

2  =  rt  I  log  tan  2  +  cos  (^  J  , 
*  Page  51. 


HISTORICAL    DEVELOPMENT   OF   PSEUDOSPHERICAL    SURFACES.  55 

where  0  is  the  obtuse  angle  between  the  axis  of  rotation  z  =  a  constant  and  the 
tangent  to  a  meritlau  curve  at  any  point,  the  equation  for  »■,  the  radius  of  a 
parallel  circle,  is 

a  .      , 

r  ^  =  sm  4>, 

where  A;-  =  =  0  for  the  three  surfaces  respectively. 

These  results  led  him  to  announce  the  theorem  that  "  a  sui'face  complementary 
to  a  surface  of  constant  negative  curvature  with  respect  to  a  system  of  geodesic 
lines  which  go  out  from  a  point  on  the  surface  is  developable  upon  a  rotation 
surface  which  has  for  its  axis  the  asymptote  and  for  its  meridian  curve,  a  cur- 
tailed tractrix,  the  ordinary  one  or  an  elongated  one,  according  as  the  point  of 
intersection  is  finite  and  real,  at  infinity,  imaginary."'  "  The  first  named 
curve,"  he  said,  "is  none  other  than  the  orthogonal  projection  of  the  tractrix 
upon  a  plane  which  goes  through  the  asymptote.  On  the  other  hand,  the  last 
named  curve  has  the  tractrix  for  its  orthogonal  projection." 

He  further  observed  that  the  deformed  parallels  of  the  surface  of  rotation  upon 
which  the  complementary  surface  is  applicable  correspond  to  the  deformed  parallels 
of  the  surface  of  rotation  into  which  the  original  surface  is  developable.  He 
proceeded  to  find  the  equations  for  surfaces  other  than  surfaces  of  rotation  that 
are  complementary  to  a  pseudosphere  with  respect  to  a  family  of  geodesic  lines 
of  each  of  the  three  kinds,  and  having  found  these  equations  he  showed  that  the 
corresponding  surfaces  are  applicable  upon  one  of  the  three  kinds  of  surfaces  of 
rotation.  Bianchi"'  also  extended  the  application  of  this  theorem  to  helicoidal 
surfaces,  and  found  that  there  are  also  three  kinds  of  helicoidal  surfaces  com- 
plementary to  a  pseudospherical  helicoid  corresponding  to  the  three  kinds  of  geo- 
desic lines  with  reference  to  which  they  may  be  developed. 

Since  the  surface  complementary  to  a  pseudospherical  surface  with  respect  to 
a  system  of  geodesic  lines  going  out  from  a  point  at  infinity  is  developable  upon 
a  pseudosphere  and  has  the  same  curvature  as  the  original  surface,  and  since  there 
are  oo'  systems  of  this  sort  on  a  surface  of  constant  negative  curvature,  Bianchi 
remarked  that  from  a  pseudospherical  surface  >S'|  an  infinite  number  of  new 
surfaces  S.,  of  the  same  curvature  may  be  derived,  and  that  from  each  surface 
Sr,  an  infinite  number  of  new  surfaces  tS^ ,  also  with  the  same  curvature,  may  be 
obtained  in  the  same  way  as  S^  is  obtained  from  >S'., ,  provided  that  a  fanaily  of 
geodesic  lines  on  S^  are  known,  and  so  on. 

6.  From  Bianchi's  surface  that  is  complementary  to  a  pseudospherical  surface 
with  reference  to  a  family  of  geodesic  lines  going  out  from  a  point  at  infinity, 
Kuen,  in  1879,  derived  by  the  repetition  of  Bianchi's  operation  the  equations  for 
a  new  pseudospherical  surface  which  he  classified  as  an  Ennejier  surface.  The 
paper  in  which  Kueu*'^  announced  these  results  was  referred  to  on  page  26. 

In  the  same  year  Lie^'"^  developed  Bianchi's  theorem  further.     In  a  paper 


66  E.    JI.    CODDINGTON. 

publishecl  in  the  Arcbiv  for  Mathematik  nnd  Naturvideuskab  he 
introduced  a  method  for  finding  by  means  of  a  quadrature  alone  the  geodesic 
lines  of  the  surface  of  centers  of  a  TF-surface,  and  especially  for  the  case  when 
the  surface  of  centers  is  composed  of  pseudospheres.  "  This  problem  for  deter- 
mining the  geodesic  lines,"  he  observed,  "is  equivalent  to  determining  the  lines 
of  curvature  on  the  TF-surface." 

He  supposed  the  TF-surface  to  be  referred  to  a  system  of  curvilinear  coordi- 
nates {x,  y)  and  expressed  one  of  its  principal  radii  of  curvature  R^  at  a  point 
and  the  coordinates  x^ ,  y^ ,  2^  of  a  point  on  the  corresponding  nappe  S^  of  its  evo- 
lute  surface  as  functions  of  these  parameters.  He  wrote  the  expression  for  the 
linear  element  on  the  nappe  S^  referred  to  a  family  of  geodesic  lines  and  their 
orthogonal  trajectories  as  parameters  and  of  curvature  —  l/«',  in  the  usual  form 

From  this  he  derived  the  equation  for  the  geodesic  lines 


dv  =  e-^''«  V{ds\-  dE\ )  =  e-^'"-  Vdx]  -f  dy\  +  dz\  -  dR\ 

and  observed  that  the  quantity  under  the  sign  of  the  radical  is  of  the  foi-m 

{X{x,y)dx  +  Y{x,y)dy]\ 

where  JTand  l^are  functions  of  x  and  y  only,  so  that  he  could  at  once  obtain 
the  integral  of  the  equation  containing  an  arbitrary  constant.  Therefore,  if  he 
had  given  a  surface  S^  with  curvature  —  l/a",  he  could  bring  its  linear  element 
in  00 '  ways  into  the  form 

ds\  =  dR\  +  e-''''"dv-  * 

referred  to  a  family  of  geodesic  lines  going  out  from  a  point  at  infinity  and  their 
orthogonal  trajectories,  and  considering  this  surface  as  the  first  nappe  of  a  sur- 
face of  centers  he  could  derive  an  involute  surface  corresponding  to  one  of  those 
infinite  systems  of  geodesic  lines.  From  this  involute  surface  he  could  obtain 
a  second  nappe  S^  of  a  surface  of  centers  and  by  the  method  just  given  deter- 
mine on  it  a  family  of  geodesic  lines  and  write  its  linear  element  in  the  form 

ds'i  =  dR^  +  e--''-"du°. 

A  repetition  with  respect  to  S„  of  the  operations  performed  on  S^  would  then 
enable  him  to  obtain  a  new  set  of  surfaces  S^  and  by  the  successive  application 
of  this  same  process  he  could  derive  oo  *  surfaces  all  with  the  same  constant 
negative  curvature  —  1/a".  In  actual  practice  he  remarked  "it  is  possible  to 
go  directly  from  one  nappe  to  a  second  without  stopping  to  obtain  the  involute 
surface." 

It  may  here  be  remarked  that  several  years  later  in  1888  Weingarten '"' 
developed  another  method  for  finding  the  lines  of  curvature  on  a  TT'^surface  and 

*  Page  12. 


HISTORICAL    DEVELOPMENT    OF    PSEUDOSPHERICAL    SURFACES.  57 

consequently  the  corresponding  geodesic  lines  on  its  surface  of  centers,  wliicli, 
according  to  Darboux,'^'' '  "''\  is  "more  precise  but  less  direct "' than  that  of 
Lie. 

After  having  developed  this  method  for  determining  the  geodesic  lines  of  a 
pseudospherical  surface  Lie  next  called  attention  to  a  method  for  transforming 
one  surface  into  another  that  had  been  discovered  by  Bonnet  many  years  before. 

Bonnet^'''"'  had  shown  that  every  surface  of  constant  mean  curvature  is 
applicable  upon  an  infinite  number  of  surfaces  of  the  same  sort  and  that  such 
a  surface  is  parallel  to  a  surface  of  constant  total  curvature  and  obtainable 
from  it  by  dilatation.  Lie  suggested  therefore  that  if  a  parallel  surface  be 
derived  from  a  pseudospherical  surface  and  transformed  into  an  infinity  of  new 
surfaces  with  constant  mean  curvature  and  each  of  these  in  its  turn  be  trans- 
formed back  into  a  pseudospherical  surface,  the  result  will  be  the  same  as  if 
Bianchi's  operation  had  been  performed  upon  the  original  surface  of  constant 
negative  curvature. 

He  showed  moreover  that  the  asymptotic  curves  of  a  surface  of  constant  curva- 
ture may  be  found  by  a  simple  integration  and  that  they  correspond  to  the 
minimal  lines  of  a  parallel  surface,  a  theorem  which  furnished  the  means  of 
obtaining  the  equation  of  transformation  as  it  is  given  by  Darboux.* 

Lie  used  for  the  linear  element  of  the  surface  referred  to  its  lines  of  curvature 
?/  =  a  constant  and  v  =  a.  constant  the  expression  due  to  Weingarten  •"  f 

and  for  the  asymptotic  lines  of  the  surface  u^  =  a  constant,  and  c,  =  a  constant 
the  corresponding  expression 


This  last  equation  he  saw  is  integrable  if 


m- 


{ inconstant). 


4>" 

The  general  integral  of  this  equation  is 

4P-  =  AK~  +  LA-, 

and  the  total  curvature  of  the  corresponding  surface  is  constant  for 

■Ri^i^  ^^  -^-4"  (i?i,  if2=  principal  radii  of  carvatnre). 
while  a  singular  integral  is 
*Daebodx,>59  §775. 
t  Page  49. 


58  E.    M.    COPDINGTOX. 

and  the  mean  curvature  of  the  corresponding  surface  is  constant  for 

^= j^  =  0      (  i^i,  i?j  =  principal  radii  of  curvature). 

He  did  not  prove  his  theorem  in  detail  nor  give  the  equations  of  transformation 
deduced  from  Bonnet's  theorem,  but  Darboux,  i^^,  ?r7s  j^  jjjg  celebrated  work  con- 
cerning surfaces,  gives  a  simple  proof  for  the  correspondence  between  the  asymp- 
totic lines  on  the  surface  with  constant  curvature  and  the  minimal  lines  on  the 
surface  with  constant  mean  curvature  and,  then,  denoting  the  linear  element  on 
the  first  surface  referred  to  its  asymptotic  lines  as  parameters  by 

ds-  =  da-  +  2  cos  codad^  +  d^' 

and  the  linear  element  of  a  parallel  minimal  surface  referred  to  its  minimal  lines 

as  parameters  by 

ds-  =  ie'^dadlS, 

where  2&)  is  the  angle  between  the  asymptotic  lines  and  also  the  angle  between  the 
minimal  lines,  he  pointed  out  that,  when  either  surface  is  transformed  into  a  new 
surface  of  the  same  kind,  the  equations  of  transformation  will  be 


n{a,/3)  =  co(^^^,a/3\ 


where  2fl  is  the  angle  between  the  asymptotic  lines  of  the  new  surface  with  con- 
stant curvature  or  the  angle  between  the  minimal  lines  of  the  new  surface  with 
constant  mean  curvature  and  a  is  a  constant. 

The  next  year  Lie'''  raised  the  question  whether  the  surfaces  obtainable  from 
one  that  is  known  by  Biauchi's  method  of  transformation  are  all  distinct  from 
each  other  or  whether  a  finite  number. of  them  are  coincident,  or  as  he  expressed 
it,  "whether  those  surfaces  of  constant  curvature  1/a",  which  are  derived  by 
the  infinitely  repeated  successive  application  of  Bianchi's  operation  from  one 
that  is  given,  must  satisfy  still  other  differential  equations  beside  the  equation 

rt  —  s'-  1 


He  answered  this  question  in  the  negative  and  his  method  of  proving  his  answer 
correct  applies  to  surfaces  of  constant  positive  curvature  as  well  as  to  those  of 
constant  negative  curvature. 

He  began  his  demonstration  by  writing  down  the  equations  which  represent 
known  characteristic  properties  of  the  nappes  of  an  evolute  surface  with  con- 
stant negative  curvature 


HISTORICAL   DEVELOPMENT    OF    PSEUDOSPHERICAL    SURFACES.  59 

(x  -  .r,  y  +  {!/-  !/, )-  +  (s  -  s, )-  =  cr, 

Pii-'^  - ^\)  +  ?i(y -yi) - ( ' -  -i)  =  0 , 

l\n  +  q^q  +  1  =  0  , 

where  (a;,  y,  :i,p^  q)  and  (.Cj,  y^,  z^,  p^,  yj  determine  an  element  on  the  first 
and  second  nappes  respectively. 

The  first  equation  shows  that  the  distance  between  corresponding  points  on 
the  nappes  is  constant,  the  last  three,  that  tangent  planes  to  the  nappes  at  cor- 
responding points  meet  at  right  angles  along  a  common  tangential  line.  In  this 
way  he  showed,  as  Biicklund  ■"  remarked,  that  the  surface  on  whose  tangent 
planes  lie  the  circles  with  constant  radius  of  a  Ribaucour  cyclic  system  and  the 
family  of  surfaces  noi-mal  to  the  circles  are  identical  respectively  with  Bianchi's 
initial  sui-face  and  the  infinite  number  of  its  complementary  surfaces,  for  these 
equations  express  analytically  the  fact  that  a  system  of  surfaces, 

are  orthogonal  at  the  point  (a;,  //,  s)  to  a  family  of  circles  with  constant  radius 
a  and  with  their  centers  lying  on  the  lines  of  curvature  of  a  surface 

»=/(-''■'  y)- 

Lie  called  his  initial  system  of  equations  the  equations  of  transformation, 
and  since  he  had  four  of  them  from  which  to  eliminate  the  five  variables,  he  saw 
that  to  every  element  [x,  y,  z,j),  q)  there  corresponds  an  infinity  of  elements 
(cCj,  y, ,  «j,  Pj,  5'j),  so  that  to  the  co-  elements  that  go  to  make  up  the  original 
surface  there  corresponds  oo'  new  elements,  and  he  proved  that  these  oo'  new 
elements  can  form  oo'  surfaces  when  the  curvature  is  constant.*  By  applying 
his  equations  for  transformation  he  was  thus  able  to  obtain  oo'  surfaces  <^,  from 
one  surface  i^and  from  these  derived  surfaces  (j)^,  oo"  new  surfaces  F.,,  among 
which  may  be  the  first  surface  F.  By  repeating  successively  this  operation  he 
finally  obtained  oc-'"  surfaces  F  and  00-'"  +  '  surfaces  (j>,  for  the  surfaces  of  the 
one  class  are  finitely  distinct  from  those  of  the  other,  but  he  had  still  to  decide 
whether  he  could  derive  all  the  pseudospherical  surfaces  in  this  way,  or  only  a 
limited  number  of  them. 

He  considered  two  surfaces,  i^and  i^,,  which  differ  so  little  that  the  one  may 
be  deformed  into  the  other  by  an  infinitesimal  transformation.  By  carefully 
working  out  the  equations  for  this  infinitesimal  transformation  he  found  three 
different  equations  for  Sp  and  Bq ,  the  increments  of  p  and  q ,  for  determining 
the  way  in  which  an  element  x,  y,  z,  p,  q  passes  into  its  next  adjacent  position. 
He  then  assumed  that  this  element  could  not  go  over  into  all  the  new  elements 

*  Cf .  page  65. 


60  E.    M.    CODDINGTON. 

but  only  into  a  certain  number  of  them  which  form  a  locus  defined  by  the  equation 

/(;o,  2/,  z,2),  5)  =  0. 

He  saw  that  this  locus  must  be  deformed  into  itself  by  the  same  operations 
which  transform  the  elements  iufinitesimally  and  that,  therefore,  it  must  satisfy 
three  equations  of  condition,  one  corresponding  to  each  of  the  three  different 
pairs  of  value  of  Sjj  and  hq .  But  from  these  same  equations  of  condition  he 
found  that  the  partial  derivatives  of  the  first  order  of  f  with  respect  to  each  of 
the  five  variables  vanish  indejiendently,  that  consequently  the  locus  f  cannot 
exist,  but  that  each  element  passes  over  in  aU  the  new  elements  and  the  given 
surface  is  deformed  by  the  equations  of  ti-ansformation  into  oc'  new  surfaces. 

In  like  manner  he  found  that  the  given  surface  can  satisfy  no  partial  differen- 
tial equation  of  the  second  or  third  order  and  accordingly  may  be  transformed 
into  00^  or  oo''  new  surfaces,  but  he  could  not  arrive  at  any  general  result  by 
this  method.  He""' next  turned  to  the  consideration  of  a  strip  on  the  given 
surface  formed  by  an  aggregation  of  successive  elements  and,  thex^efore,  trans- 
formable into  00 '  new  strips.  He  proved  by  the  actual  application  of  the 
equations  for  a  Bianchi  transformation  that,  if  the  curve  C,  formed  by  the 
points  of  all  the  surface  elements  along  a  strij)  is  an  asymptotic  curve  it  may  be 
deformed  into  oo'  new  asymptotic  curves  JZ,  that  the  arc  length  of  each  new 
curve  A' is  equal  to  the  corresponding  arc  length  of  C  and  that  the  curvature 
l/i?j  of  each  new  curve  is  related  to  the  curvature  1/i?  of  C  by  the  equation 

a         a  . 

-^=-^-2sm.. 

where  p  is  the  angle  that  a  line  joining  a  point  on  the  one  curve  to  a  corre- 
sponding point  on  the  other  curve  makes  with  the  tangent  to  either  curve  at  the 
point  where  the  curve  is  met  by  the  line.  He  derived  oo"  new  asymptotic 
curves  C,  from  the  curves  A';  by  a  third  repetition  of  the  oj)eration  he  obtained 
00*  new  curves  A',  and  so  on,  so  that  the  problem  as  in  the  case  of  surfaces 
resolves  itself  into  the  question,  is  there  any  limit  to  the  number  of  asymptotic 
curves  that  are  thus  derived?  Reasoning  in  the  same  way  as  for  the  infinitesi- 
mal transformation  of  surfaces,  he  found  that  the  number  of  asymptotic  curves 
that  can  be  derived  from  one  that  is  known  will  be  reduced,  only,  if  these  derived 
curves  can  satisfy  an  ordinary  differential  equation.  Denoting  a/Ji  by  v  and 
a/H^  by  i\  he  wrote  the  equation  connecting  these  values  in  the  form 

V  =  i\  —  2  sin  V 
and  the  equation  for  v  in  the  form 

dv 

c  ^-  =  —  t),  -f  sm  V . 
as  ' 


HISTORICAL   DEVELOPMENT    OF   FSEUDOSPHERICAL    SURFACES.  61 

He  then  proved  that  such  an  equation  as 

cannot  exist,  so  that  any  asymptotic  curve  corresponding  to  u  =  a  constant  can 
be  transformed  into  at  least  oo'  new  asymptotic  curves.  He  proved  that  there 
is  no  relation  between  v  and  its  derivatives  of  the  second  or  third  order  with 
respect  to  s  nor,  indeed,  between  v  and  its  derivatives  of  any  order  for  on 
account  of  the  form  of  the  equations  for  the  increment  of  i\  Sv  and  Sv"  he  could 
write  down  by  analogy  the  equation  for  Bv"  and  then  show  that  the  one  for 
Bv"  + '  is  exactly  similar.  He  thus  showed  that  there  is  no  limit  to  the  number 
of  asymptotic  lines  that  can  be  obtained  from  a  given  one  by  the  equations  of 
transformation. 

He  then  turned  back  to  the  case  of  the  surfaces  and,  by  means  of  his  new 
results,  increased  the  number  of  surfaces  that  can  be  derived  from  a  known  one 
which  passes  through  two  intersecting  asymptotic  curves  from  oo^  to  oo*  thus 
establishing  his  theorem.* 

Lie  proved  that  there  is  not  only  a  correspondence  between  the  asymptotic 
lines  on  a  transformed  surface  with  those  on  the  initial  surface  but  also  one 
between  their  lines  of  curvature,  for  since,  according  to  Dini's  discovery,  the 
asymptotic  lines  of  a  surface  divide  it  into  lozenges,  a  net-work  of  lozenges  on 
one  surface  S  is  transformable  into  a  net-work  of  lozenges  on  each  of  the  derived 
surfaces,  and  the  lines  of  curvature  which  are  their  diagonals  pass  over  into 
lines  of  curvature. 

During  these  same  years  from  1879  to  1882  while  Bianchi  and  Lie  were 
making  their  important  investigations  on  the  method  of  obtaining  new  surfaces, 
of  constant  curvature  from  a  given  one,  Biiekliind  '"■  "  was  publishing  the  results 
of  his  studies  on  the  transformation  of  surfaces  in  successive  volumes  of  the 
Mathematische  Annalen  and  the  discoveries  of  Bianchi  and  Lie  were  made 
just  at  a  time  when  Biicklund  could  use  them  as  examples  to  illustrate  his 
theorems. 

Among  other  propositions  Biicklund'"  considered  the  question  whether  two 
surfaces  may  be  transformed  into  each  other  when  the  relation  between  them  is 
of  such  a  nature  that  it  is  defined  by  four  arbitrary  partial  differential  equations 
of  the  first  order. 

He  denoted  the  two  surfaces  by 

z  =4>{x,  y)         and  z'  =f(x',  y'), 

and  using ^>,  ^,  ?■,  s,  ^  to  denote  the  partial  derivatives  of  the  first  and  .second 
order  of  z  with  respect  to  x  and  y  as  is  customary,  andp',  q' ,  r',  s' ,  t'  to  denote 

*  Bianchi,  ■»  ?  247. 


62 


E.    M.    CODDINGTON. 


the  jmrtial  derivatives  of  the  first  and  second  order  of  z   with  respect  to  x   and 
y'  he  wrote  the  four  partial  differential  equations  in  the  form 


F,{x, 

y. 

8,iJ, 

q,x 

.y\ 

z 

^P' 

l')  = 

0. 

F,{ 

)  = 

0. 

F,{ 

)  = 

0. 

K{ 

)  = 

0. 

He  then  proceeded  to  find  under  what  condition  the  surface  whose  equation  is 

z  =  4,{x,  y) 
may  be  transformed  into  the  surface  whose  equation  is 

z   =f(x',  y') 

by  means  of  these  equations,  he  first  substituted  in  the  equations  i^^  =  0  and 
i^j  =  0  the  values  of  z,p,q  expressed  as  functions  of  x  and  y .  He  then  solved 
the  resulting  equations  for  x  and  y  expressing  them  in  terms  of  the  accented 
variables  only.  By  means  of  these  results  he  could,  by  making  the  proper 
substitutions,  reduce  the  last  two  equations 

i^3  =  0  and  F^  =  0 

to  equations  containing  a;',  y',  s',  p  ^  q    only,  in  which  case  he  denoted  them  by 

i^;  =  0  and  i^;  =  0 . 

He  could  then  obtain  the  function  z   by  means  of  these  equations  provided  that 
they  are  compatible.     The  equation  of  condition  which  must  be  satisfied  by  z 
when  the  two  equations 

are  compatible  may  be  obtained  by  first  taking  the  total  derivatives  of  each  of 
these  equations,  which  are  also  equal  to  zero,  then  solving  the  resulting  equations 
for  dp   and  dq    so  that 


dF'.    dF: 

dF'^        ,dF'.    dF'. 

3                 3 

dx'   ^i      dz'  '    dq' 

dq    '    dp 

dp'  = 

dx'  + 

dF',,    dF\ 

i. 

dF'^        ,dF'^    dF'^ 

-~  +r>  --1-  ,  ^--^ 
dx         '■      cz         cq 

dq    '    dp' 

dF'^    dF'^l 

dq  = 

dF'.         ,dF'^    dF'^ 

3      1          ' 3 i 

dx     ^^     dz    '    dp' 

dx'  + 

dF',    dF\ 

dF\        ,dF'^    dF'^ 

4                   4 

dq    '    dp' 

dx'    '^^    dz'  '    dp' 

dF',  ,dF',  dF', 

cy  ^     cz  dq 

dF\  ,dF\  dF', 

cy  ^     cz  cq 


dy' 


+  9^, 


dz'  '    dj)' 


dF',        ,dF',    dF\ 
dy         ^    cz        cp 


dy' 


dy' 


and  finally  setting  the  coefficient  of  dy'  in  the  first  of  these  last  equations  equal 


HISTORICAL    DEVELOPMENT    OF   PSEUDOSPHERICAL    SURFACES.  63 

to  the  coefficient  of  dx'  in  the  second  equation  from  which  results  the  equation 
{dF;AdF^_/dF^\dF:       (dF^\  BF:  _(dF\\  dF[  _ 

\  dx  )  dp'      \  dx  )  dp'  +  V  ~(iy'  )  ^q      \  H  )  ^q'  ~ 


where 


/dF'.\      dF'.         ,dF'.         ,dF'.        ,dF'. 


m- 


dz'     ^        dp'    ^^     Bq' 

{i  =  3,  4). 
dF'.  dF'.         ,cF'.  dF'. 

-dy'     +^-d7    +'^dp^+'^ 


This  equation  he  represented  by  the  bracket  \_F'^F'^'\f,,,p.=  0  and  when  instead 
of  F'^  and  F'^  he  introduced  their  equivalent  values  in  terms  of  the  unaccented 
variable  the  eciuation  became 

IF'.F',]  =  [i^3^J=.v  +  '2^[»^i^J  +^'[2/^J  +^^^[^3^] 


or  finally 


dF.^^    ^       VdF,dF.      dF.dF.l^      ,       „ 


[F:F,]=iU)[F,F,],^,^,+  {i2)[F,F,],,,,.+  (2S)[F,FJ,,^, 

+  (12)  [F^F,],,^,  +  (13)  [F,F.,],^,,.  +  (14)  [F,F^],^,„  =  0, 
where 

,       ,       /dF  \  (dF  \       (dF  \  (dF  \ 

He  had.  therefore,  three  equations 

containing  the  accented  variables  only,  which  will  determine  a  surface  vJ  = /"(.r',  y') 
and  only  one  surface,  provided  that  these  equations  are  in  involution,  a  condition 
which  he  represented  in  the  usual  manner  by  the  equations 

He  moreover  showed  that  the  function  z  =/'(.t',  j/' )  will  satisfy  two  partial 
differential  equations  of  the  third  order  obtained  by  eliminating  cc',  ?/',  «',  p  ^  q 
from  the  four  equations  of  transformation  and  from  the  equations  of  condition 
\_F'^F'^'\  =  0.     He  remarked  that  an  excejition  to  this  theorem  occurs  when 
z'  does  not  appear  in  the  equation 

[F',F',]  =  0 

and  that  then  x^  surfaces  2'  =f(^x',y')  will  correspond  to  one  surface 2  =  ^(x,?/), 

*z'x'p'   written  after   the   limcket  signifies  tliat  Fi  is  differentiated  witb  respect  to  the 
accented  variables  only. 


64  E.    M.    CODDINGTON. 

for,  in  that  case,  instead  of  two  partial  differential  equations  of  the  third  order 
for  z  there  will  he  one  single  partial  differential  equation  of  the  second  order, 
and  if  an  integral  of  this  equation  he  substituted  for  s  in  the  equation  of  trans- 
formation, the  quantities  x\  y' ip' ■,  q  can  be  expressed  in  terms  of  x,  y,  z  so  that 
the  function  .-'  will  be  determined  by  an  equation  of  the  form 

dz  =  A{x,  y,  z')dx  +  B(x,  y,  z')dy. 

The  integral  of  this  last  equation  will  contain  an  arbitrary  constant  which  proves 
the  theorem  that  there  are  oo^  sui'faces  z'  =y'(.c',  y' )  corresponding  to  one  sur- 
face z  =  4>{x,  y). 

BKcklund"''  saw  that  a  surface  transformation  of  this  nature  occurs  in  Bianchi's 
problem  for  deriving  a  surface  complementary  to  a  known  surface  of  constant 
negative  curvature.     He  considered  the  two  surfaces  defined  by 

s  =  c^(x,2/)  z'=f{x',y') 

as  the  two  nappes  of  the  evolute  surface  whose  radii  of  curvature  are  connected 
by  the  relation 

The  two  relations  existing  between  the  two  nappes,  that  the  distance  between 
corresponding  points  is  a  constant  a  and  that  their  tangent  planes  at  correspond- 
ing points  must  meet  at  a  right  angle  along  the  common  tangent,  gave  him  his 
four  equations  of  transformation 

F,  =  p{x'  -  X)  +  q(y'-y)-(z'-z)  =  0 

F,  =p'  {x  -x)  +  q  {y'  —y)—{z'-z)  =  0 

F.^=l-{-pj/  +qq'  =  0 

F^  =  ix-x'y  +  {y-y'y  +  {z-z'f-(r=0. 

and  his  equation  of  condition  took  the  form 

(rt  -  s"-)  +  «-(l  +  2r  +  q-y-  =  0 , 

since  the  expressions  [F^F^],^,^„  [F,F^],^,^„  [F,F,],^,^„  [F,FJ,,^,  all 

become  equal  to  zero.  He  saw  from  this  equation  and  from  a  similar  one  for  z, 
since  the  equations  of  transformation  are  symmetrical  with  respect  to  the 
accented  and  unaccented  variables,  that  both  surfaces  are  of  constant  negative 
curvature  —  l/a'  and  that  since  z'  does  not  appear  in  the  equation  of  condition 
that  there  correspond  an  infinity  of  surfaces  z' =J\x' ,  y' )  to  evei-y  surface 
z  =  (^(x,  y). 

In  1884  BKcklund  ^' wrote  an  important  paper  that  deals  exclusively  with 
pseudospherical    surfaces.     In    this    paper,    published  in    the    Lund's    Uui- 


HISTORICAL   DEVELOPMENT    OF   PSEUDOSPnERICAL    SURFACES.  65 

versifcets  Arsskrift  and  entitled  "Concerning- Surfaces  with  Constant  Nega- 
tive Curvature,"  lie  first  reviewed  the  contributions  made  to  the  theory  of  the 
transformation  of  the  pseudosjjherical  surfaces  by  Bianchi,  Ribaucour  and  Lie, 
pointing  out  the  close  connection  between  the  theories  of  Bianchi  and  those  of 
Ribaucour,  he  then  extended  Bianchi's  theorem  to  fit  a  more  general  case,  namely, 
when  the  given  surface  and  the  derived  surface  are  not  the  nappes  of  an  evolute 
surface  but  are  so  related  to  each  other  that  planes  tangent  to  them  at  corre- 
sponding points  cut  each  other  at  a  constant  angle,  but  not  at  right  angles,  and 
the  distance  between  two  corresponding  points  is  constant.  He  expressed  this 
condition  by  leaving  the  first  three  equations  of  transformation  unaltered  and 
writino;  i^,  =  0  in  the  form 


^4  =  1+  pp'  +  n  - ^^{ v  1  +  f  +  r){  1'  1  +  p"  +  ?")  =  0 

where  K  is  the  cosine  of  the  angle  formed  by  the  two  tangent  planes  and  is  a 
constant. 

He  then  found  that  the  equation  for  z  becomes 

rt  -s-=  -  -— r-  (1  +  F  +  r)'' 

and  that  a  like  one  exists  for  z' ,  so  that  in  the  general  case  also  both  surfaces 
have  constant  negative  curvature.  By  putting  for  (1— A'')/(r  a  constant 
Ijm-  and  letting  a  and  TT  vary,  he  obtained  an  infinity  of  equations  of  transfor- 
mation for  surfaces  only  whose  curvature  is  —  \jm-  and  in  particular  those  for 
Bianchi's  complementary  transformation  when  Il  =  0  and  m  =  a. 

He  made  a  complete  study  of  this  general  method  of  transformation.     First 

he  remarked  that  the  set  of  equations 

f'(x)-y!r(x) 
«=/(x),y  =  c^(:«),         I>  =  ir(x),  q='   "^  i'{x) 

determine  a  curve  on  the  surface  together  with  the  direction  of  the  tangent  plane 
to  the  surface  along  that  curve  for  successive  values  of  x,  that  is,  they  determine 
a  strip  of  the  surface.  Then  recalling  Cauchy's  theories  he  observed  that  if  .r,  y 
and  a  are  the  coordinates  of  an  arbitrary  point  in  the  strip  and  cb^,  y^,  z^  the  coordi- 
nates of  its  initial  point,  a  surface  passing  through  this  strip  and  satisfying  a 
known  differential  equation  may  be  defined  by  a  convergent  Taylor's  series  in 
terms  of  {x  —  x^,  {i/  —  y^)  where  the  singular  points  of  the  surface  are  not  con- 
sidered, thus 

«  -  '^0  =Po{^'  -  ^o)  +  loiy  -  2/o) +H'-o(a'  -  ^o) '+  2s„(a^  -  Xo)(y  -  y,)  +  t,{y  -  y.f} 

+  3j  {u,{x- xj  -f  3r„(x - o:J{y- y,)  +  3«.,(.-.3 - .r„)(.y -  y,f  +  o>^X>J  -  %Ti 

+  •••• 


66 


E.    M.    CODDIXGTOX. 


For  the  surfaces  under  discussion  he  obtained  the  values  of  the  coefficients  r,  s, 
etc.,  from  the  equations 

dp  =  nix  +  sdi/,         dq  =  sdx  +  tdy , 

SO  that  t  is  determined  by  the  equation 

dpdx  +  dqdy 

In  general  only  one  surface  can  be  found  passing  through  the  strip,  but  when 
the  value  of  t  is  indeterminate,  that  is,  when 

dpdx  +  dqdy  =  0  and  dq- 2(1+  P'  +  q")'dx-  =  0 , 

Backhmd  saw  that  there  are  an  infinity  of  surfaces  having  contact  of  the  first 
order  along  the  lines  defined  by  those  equations  and  that  these  lines  are  the 
characteristic  curves  of  the  integral  surface.  Since,  the  first  of  these  equations 
shows  that  each  curve  may  have  for  its  plane  of  osculation  at  every  point  the 
tangent  plane  to  the  surface  on  which  it  lies  at  that  point  and  the  second  equa- 
tion shows  that  the  torsion  of  the  curve  is  constant,  Backlund  thus  proved  that 
the  characteristic  curves  of  surfaces  of  negative  constant  curvature  are  asymptotic 
curves. 

He,  then,  demonstrated  geometrically  that  the  guiding  curve  of  every  strip  )•', 
derived  from  a  strip  r  on  the  original  surface  S,  by  means  of  the  set  of  equations 
of  transformation  satisfies  a  partial  differential  equation  of  the  Kiccati  type  and 
that,  consequently,  every  strip  ;•'  corresponds  to  a  solution  of  such  an  equation. 

This  final  result  may  then  be  stated  as  follows  :  If  the  surface  S  is  known,  all 
the  surfaces  S'  may  be  derived  from  it,  for  every  strip  r  on  S  passes  over  into 
an  infinity  of  strips  »■',  one  on  each  surface  S' ,  by  means  of  the  equations  of 
transformation,  therefore  each  derived  surface  S'  corresponds  to  the  solution  of 
a  Riccati  equation  and  when  one  such  surface  is  determined,  all  the  others  may 
be  found  by  quadrature  alone,  since  that  is  the  only  operation  required  to  obtain 
all  the  solutions  of  a  Riccati  equation  when  one  is  known.  Backlund  has  jjroved 
geometrically  that  the  asymptotic  curves  on  the  new  surfaces  *S"  are  the  deformed 
asymptotic  curves  of  the  original  surface  S  and  that  they  also  satisfy  an  equation 
of  the  Riccati  form. 

In  1883  and  1884,  Biauchi  ^''  *»•  '"■  ''• »'  published  his  investigations  of  Ribau- 
cour's  propositions  concerning  a  system  of  surfaces  which  have  a  family  of  00^ 
circles  for  their  orthogonal  ti-ajectories.     Biicklund  ^"  had  already  observed  that, 


HISTORICAL   DEVELOPMENT   OF    rSEUDOSPHERICAL   SURFACES.  67 

when  the  circles  all  have  the  same  constant  radius  aiul  lie  in  the  tangent  planes  of 
a  known  surface,  this  known  surface  and  the  surfaces  orthogonal  to  the  circles  are 
identical  with  a  pseudosphere  and  the  oo'  surfaces  derived  from  it  by  means  of 
a  complementary  transformation  with  respect  to  a  family  of  geodesic  lines  on  it 
that  go  out  from  a  point  at  infinity.  Bianchi  gave  an  exact  proof  of  tlie  iden- 
tity of  the  two  families  of  surfaces  by  establishing  the  theorem  that  a  surface 
orthogonal  to  a  family  of  oo'  circles,  can  be  regarded  as  the  nappe  of  the  evolute 
surface  of  a  TF-surface,  provided  that  the  line  of  intersection  of  the  plane  of 
every  circle  with  the  planes  tangent  to  the  ortiiogonal  surface  at  its  point  of  con- 
tact with  that  circle  envelopes  geodesic  lines  on  the  surface  and  by  showing  that 
those  enveloped  curves  are  geodesic  lines  when  the  radius  of  the  circles  is  always 
the  same.     Bianclii's  construction  of  a  cyclic  system  of  surfaces  is  as  follows : 

Let  /S'j*  be  a  surface  orthogonal  to  a  family  of  circles.  Let  these  circles  lie 
on  the  tangent  planes  of  a  second  surface  S.,,  and  let  the  points  of  tangency  of 
those  planes  with  the  surface  S^  be  the  centers  of  the  circles.  Let  u  =  a  con- 
stant and  V  =  a  constant  denote  the  lines  of  curvature  of  the  surface  S.^  and  let 
6  be  the  angle  that  a  radius  of  the  circle,  mn,  drawn  to  meet  an  orthogonal 
surface  S^  at  m  makes  at  its  center  n  with  the  line  of  curvature  d  =  a  constant 
passing  through  that  point.  The  radius  of  the  circle  mn  being  tangent  to  the 
orthogonal  surface  S^  must  lie  in  the  tangent  plane  at  m  and  is  the  line  of  inter- 
section of  the  plane  of  the  circle  with  the  corresponding  tangent  plane  of  the 
orthogonal  surface  S^.  Let  u'  and  v'  be  the  lines  of  curvature  of  S^.  Let  (^ 
be  the  angle  which  this  line  of  intersection  makes  with  the  tangent  to  the  line 
of  curvature  of  ,S'| ,  u'  =  a  constant,  at  the  point  of  contact  m.  When  the  sur- 
face S^  is  regarded  as  known,  each  orthogonal  su-rface  S^  corresponds  to  a  value 
of  6.  When  an  orthogonal  surface  ^S",  is  regarded  as  known,  each  circle  is 
determined  by  its  radius  and  the  value  of  (/>  to  which  it  corresponds. 

Bianchi  *^  denoted  by  <J>  =  a  constant,  the  curves  on  the  orthogonal  surface  S^ 
which  are  the  orthogonal  trajectories  of  the  curves  on  that  surface  that  are 
enveloped  by  the  lines  of  intersection  of  the  planes  of  the  circles  with  the  corre- 
sponding tangent  planes  of  the  surface  ^S'^.  When  the  linear  element  of  this 
surface  S^  referred  to  its  lines  of  curvature  as  parameters  assumes  the  form 

dSl  =  Edu-  +  Gdv-, 

he  showed  that  <I>  must  be  a  solution  of  the  differential  equation 

d'^       d^   d^         1    d\/'E  d^         1     dVG  d^ 
dudv  ~  du  '  dv        -^^    dv       du       •^'Q    du       dv' 

which,  if  log  Z  is  written  in  place  of  <&,  reduces  to  an  equation  for  Z  identical 
with  that  given  by  Kibaucour. 

*Daeboux,>39  I  804;  Bianchi,""  §  179. 


68  E.    M.    CODDINGTON. 

He  also  proved  Eibaucour's  statement,  that  it  is  necessary  to  be  able  to 
integrate  this  equation  in  order  to  find  all  the  C3"clical  systems  of  which  the  sur- 
face /S',  forms  a  part,  for  he  derived  for  li  and  (/>,  the  functions  by  which  a 
circle  is  determined,  the  following  expressions 

1  1    d^  1    ccj) 

-^„  =  A.  $,*  cos  <l>  =  B  — ^^^  ,         sin  4>  =  H  — .--  ^^, 

which  can  be  found  when  the  surface  S^  and  a  value  of  (j)  ai-e  known. 

From  his  equations  for  expressing  the  condition  that  the  circles  are  orthogonal 
to  a  surface  S^,  Bianchi  was  able  to  show  that  when  the  circles  have  all  the 
same  constant  radius  i?,  the  surface  /S',  as  well  as  the  surface  S,,,  on  whose 
tangent  planes  the  circles  lie,  wiU  both  have  constant  negative  curvature 
—  l/M'.     In  that  case  he  saw  that 

A,  <I>  =  a  constant, 

or  that  ^  =  a  constant  ai-e  geodesic  parallel  circles  and  that  the  curves  enveloped 
by  the  lines  of  intersection  of  the  planes  of  the  circles  with  the  corresponding 
tangent  planes  of  the  orthogonal  surface  /S*,  and  which  are  the  orthogonal  tra- 
jectories of  ^  =  a  constant  will  be  geodesic  lines.  Moreover,  he  found  that 
the  geodesic  curvature,  1//3,  of  the  curves  <I>  =  a  constant  is  equal  to  1/^  so 
that  when  H  is  constant  they  are  the  deformed  horicj'cles  of  the  pseudosphere 
on  which  the  surface  S^  of  curvature  —  1/i?-  is  applicable.  The  fact  that  the 
curves  on  the  surface  aS,  that  are  enveloped  by  the  lines  of  intersection  of  the 
planes  of  the  circles  with  the  corresponding  tangent  planes  to  the  surface  S^  are 
geodesic  lines  was  the  only  condition  Beltrami  required  in  order  to  prove  that 
the  surfaces  S^  and  S.-^  form  the  nappes  of  an  evolute  surface,  for  these  lines  of 
intersection,  being  tangent  to  the  surface  S^  along  geodesic  lines,  may  be  regarded 
as  the  rays  of  a  normal  congruence  of  which  the  surfaces  S^  and  S.,  are  the 
focal  surfaces. 

It  is  not  necessary  to  give  in  detail  the  equations  and  theorems  by  means  of 
which  Bianchi  proved  Eibaucour's  propositions,  that  if  a  system  of  co  ^  circles 
are  orthogonal  to  three  surfaces  they  wiU  be  orthogonal  to  oo  ^  surfaces  and  the 
theorems  relating  to  the  triply  orthogonal  systems  to  which  these  surfaces 
belong,  but  it  is  important  to  consider  a  proof  given  by  Darboux*'  in  1883  for 
the  establishment  of  the  theorem  regarding  the  existence  of  this  triply  ortho- 
gonal system,  for  in  that  connection  Darboux"'''''^''  developed  for  the  first 
time  the  now  weU  known  set  of  equations  for  performing  a  complementary  trans- 
formation. He  regarded  as  known  the  surface  S„  of  curvature  —  1  on  whose 
tangent  planes  lie  the  circles  of  the  system.     He  chose  the  lines  of  curvature 


1    /d^\-        1  /c'i\^ 
*^'*  =  G(a;J   +^(,-):  BIANCHI -§§35,  86. 


HISTORICAL   DEVELOPMENT    OF   PSEUDOSPHERICAL    SURFACES.  09 

of  this  surface  for  its  lines  of  reference  and  wrote  the  linear  element  in  the  form 

ds'  =  COS"  Q)du'  +  sin-  <odv', 
where  (o  satisfies  the  characteristic  equation 

zr—iT ^r-,,   =  Sin  (U  COS  Cl) 

and  is  half  the  angle  between  the  asymptotic  curves,  ?«  +  v  and  u  —  v.*  He 
referred  to  both  Ribaucour  and  Bianchi,  and  using  the  notation  of  the  latter, 
denoted  by  6 ,  the  angle  that  the  line  v  =  a  constant  makes  with  the  radius  7nn 
of  a  circle  in  a  plane  tangent  to  S^  at  m .  He  saw  that  the  coordinates  of  m 
relative  to  the  tetrahedron  at  ?«  are 

cos  0 ,         sin  6 ,         0 , 

and  expressing  the  condition  that  this  line   nui  whose  direction  cosines  with 

respect  to  the  tetrahedron's  axes  are 

^  d CO  d(o\ 

—  sin  6d6  +  cos  (odit  —  (   ^^  d^l  +  ^-  I  sin  0 , 
\  cv  cu  )  ' 

QdQ  +  sin  wdv  +  (  -.^  du  +  ^  -  dv  )  cos  ^ . 
\  cv  cu       J 

cos  o)  sin  ddv  —  sin  co  cos  ddu 

should  be  perpendicular  to  the  tangent  to  the  circle  at  7i,  whose  direction 
cosines  are 

—  sin  6,         cos  0,         0 , 
he  obtained  the  equation 

dco  do) 

da  +  -^  du  +  p-  (Z-y  —  sin  0  cos  oofZw  +  sin  oj  cos  Bdv  =  0  , 

and  consequently  the  equations 

dO        dco        .     ^  de       dco 

^ — h  ^T"  =  sm  p  cos  6) ,         -= — \-  -—-  =  —  cos  ^  sin  oj , 

cu        cv  cv        Cll 

which  are  consistent  when  the  equation  for  co  is  satisfied. 

He  further  observed  that  each  solution  for  0  contains  an  arbitrary  constant  a 
so  that  there  may  be  an  infinity  of  surfaces  *S'j.  He  considered  0  as  a  function 
of  u,  V  and  a,  and  wrote  for  dO , 

dd  ^         B0  ^        de  ^ 
^r-  du  +  ^—  dv  +  ^r—  rfa   • 
cu  cv  ca 

*  P.  28. 


cos 


70  E.    M.    CODDINGTON. 

in  the  expressions  for  the  displacement  of  m ,  and  brought  them  to  the  form 

cos  6  {  cos  w  cos  6dxt  +  sin  w  sin  6dv )  —  sinO  ^cla, 

^  'da 

sin  6  ( cos  ft)  cos  6du  +  sin  cd  sin  ddv )  +  cos  6^  da, 

cos  (o  sin  Odv  —  sin  co  cos  0du , 

which  give  for  the  displacements  of  ii  in  space 

/  d0  \- 
ds'^  =  cos-  Odv^  +  sin-  6dv-  +  [  p-  )  f^«  > 

a  formula  which  demonstrates  the  existence    of  the  triply  orthogonal  system. 
Bianchi"'  obtained  a  like  set  of  equations  for  representing  a  Biicklund  trans- 
formation.    Employing  the  same  expression  for  the  linear  element  of  the  initial 
surface  referred  to  its  lines  of  curvature  as  Darboux  had  used, 

ds-  =  COS"  o)du^  +  sin^  ccdv', 

and  denoting  by  a-  the  complement  of  the  angle  between  the  tangent  planes  at 
corresponding  points  of  this  surface  and  a  derived  surface,  he  first  wrote  these 
equations  in  the  form  : 

89      dco      sin  6  cos  to  -\-  sin  a  cos  6  sin  to 
du      dv  ~  H  cos  a-  ' 

36      c(o  cos  6  sin  co  -\-  sin  cr  sin  6  cos  w 

dv       du  It  cos  cr 

and,  by  using  asymptotic  lines  on  the  initial  surface  for  parameters  instead  of 
lines  of  curvature,  reduced  them  to  the  simpler  form 

d(6  —  a)       1  -f  sin  o-   . 

-A- >  =  ^ sin  (^  -f  o)), 

CMj  M  COS  <T  ^  ' 

a(^+  <b)       1  —  sino- 

-^ =     T> sni  ( ^  —  ft) ) , 

dU|  It  cos  a         ^  ' 

He  saw  that  these  equations  are  compatible  if  the  curvature  of  the  initial  surface 
is  constant  and  negative,  and  that  they  form  a  Riccati  equation  for  tan  ^/2  such 
as  Biicklund  had  obtained  previously. 

Bianchi  represented  a  Biicklund  transformation  by  B^  and  his  own,  or  the 
complementary  transformation,  when  o-  =  0,  by  jB,,.  Later  he  denoted  a  Lie 
transformation  in  which,  retaining  the  previous  notation. 


0(w,  v)  =a)f  rnf,       \ 


*  Page  58. 


HISTORICAL    DEVELOPMENT    OF    PSEUDOSPHERICAL   SURFACES.  71 

by  L,  and  wrote 

1  4-  sin  a  11  —  sin  cr 

a  = —  and         ~  = . 

cos  <T  a  cos  a 

He  ""••  ''• '"'"  found  that  a  Biicklund  transformation  is  a  combination  of  a  Lie 
transformation  and  a  complementary  transformation,  or  that 

where  the  negative  exponent  denotes  an  inverse  operation. 

Biicklund's  idea  of  a  constant  angle,  not  a  right  angle,  between  the  tangent 
planes  at  corresponding  points  on  the  two  surfaces  /S,  and  /S,  that  are  deformable 
into  each  other,  apparently  recalled  to  Bianchi's  mind  Rummer's  treatise  on  ray 
systems.  He'"  asked  himself  the  question,  can  two  pseudospherical  surfaces 
developable  into  each  other  by  a  Backlund  transformation  form  the  two  focal 
surfaces  of  a  congruence  of  right  lines  ? 

Now  Kummer  has  shown  that  if  7  denotes  the  angle  between  the  tangent 
planes  to  the  focal  surfaces  respectively  that  pass  through  a  common  ray, 

where  Id  is  the  distance  between  the  limiting  points  of  the  ray  and  2S  is  the  dis- 
tance between  its  foci.  Bianchi,'-'  observing  that  when  h  and  7  are  constant,  d 
must  be  constant  also,  considered  a  ray  system  subject  to  these  conditions  and 
let  >S|  and  S^  denote  its  two  focal  surfaces,  that  is,  S^  and  S^  denote  the  two  focal 
surfaces  of  a  ray  system  which  is  characterized  by  the  property,  that  the  distance 
between  the  limiting  points  on  every  ray  is  equal  to  a  constant  R  and  the  dis- 
tance between  the  points  of  intersection  of  every  ray  with  the  focal  surfaces  is 
also  a  constant  and  equal  to  It  cos  cr,  where  (7r/2  —  <j)  is  the  constant  angle 
between  planes  tangent  at  corresponding  points  to  the  focal  surfaces. 

He"'  took  one  of  the  focal  surfaces  for  the  surface  of  reference  and  calling 
the  curves  on  it  which  are  enveloped  by  the  rays,  "  caustics,"  chose  them  and 
their  orthogonal  trajectories  for  the  coordinate  lines  n  and  v.  He  was  then 
able  by  means  of  Kummer's  equations  for  the  abscissa?  of  points  on  the  focal 
surface  and  for  the  abscissse  of  limiting  points  on  the  rays  to  prove,  first,  that 
"  the  inclination  of  the  principal  normal  of  any  caustic  of  the  focal  surface 
to  the  tangent  plane  to  the  surface  is  equal  to  the  mutual  inclination  of  the 
tangent  planes  to  the  focal  surfaces  which  pass  through  the  tangent  to  the 
caustic,"  and,  second,  that  the  two  focal  surfaces  have  their  curvature  constant, 
negative  and  equal  to  the  reciprocal  of  the  square  of  the  distance  of  the  limiting 
points  R.  He  gave  the  name  pseudospherical  congruence  to  a  ray  system  wliose 
focal  surfaces  are  j)seudosi)herical  surfaces  and  proved  that  to  any  pseudosiiherical 


72  E.    M.    CODDINGTON. 

surface  *Sj  there  belongs  a  single  infinity  of  pseudospherical  congruences  for 
which  S^  is  the  common  focal  surface. 

In  order  to  answer  his  original  question  and  demonstrate  that  two  focal  sur- 
faces of  a  pseudospherical  congTuenee  are  developable  into  each  other  by  a 
Bficklund  transformation,  he  regarded  one  of  these  surfaces  /S^  as  known  and 
defined  by  the  equation 

a=2w 

^:^-^-  =  cos  2fU, 

cvcv 

where  2&)  =  the  angle  between  its  asymptotic  curves,  u  =  a  constant  and  t)  =  a 
constant,  and  denoted  by  6  the  angle  which  a  ray,  touching  this  surface  at  any 
point  I^  and  going  out  in  the  direction  of  the  other  focal  surface  S^  which  is  to 
be  determined,  makes  with  a  line  of  curvature  of  S^  passing  through  f.  He 
then  proved  that  S^  and  >S\  are  connected  by  the  operations  for  a  Biicklund 
transformation  so  that  the  one  surface  S^  may  be  transformed  into  the  other 
surface  *S',  by  this  method,  that  the  angle  6  has  the  same  significance  for  >S'j  as 
(I)  has  for  S.^  and  that  the  linear  element  of  <S,  may  be  written 

ds'  =  sin"  6chr  +  cos"  ddv" , 

when  referred  to  its  lines  of  curvature. 

This  theorem  enabled  him  to  construct  a  geometrical  method  of  performing 
Backlund's  transformation  analogous  to  his  own  early  method  for  deriving  a 
complementary  surface  from  one  arbitrarily  chosen,  for,  he  had  only  to  select  on 
a  pseiidospherical  surface  of  any  curvature  —  l/H'  a  family  of  curves  whose 
principal  normals  make  a  constant  angle  (7r/2  —  a)  with  the  tangent  planes  to 
the  surface  and  cut  off  on  the  tangents  to  these  curves  a  constant  length  equal 
to  JR  cos  a,  then  the  locus  of  the  extremes  will  be  the  required  surface.* 

As  a  special  illustration  of  a  Biicklund  transformation,  he  investigated  the  case 
when  the  surface  derived  by  that  method  from  a  known  surface  degenerates  into 
a  straight  line.  Eepresenting  the  linear  element  on  the  derived  surface  referred 
to  its  lines  of  curvature  by 

ds-  =  sin- 6dii^  +  cos^  6dv-, 

he  expressed  the  condition  for  a  straight  line  by  putting 

sin  ^  =  0  or  cos  ^  =  0, 

and  using  the  first  expression,  reduced  his  former  equations  of  transformation  to 

dco  sin  ft)         do)      tan  o-  sin  (o 


cu  M  cos  cr'      Bv  H 

whose  common  solution  is 
*  Page  47. 


HISTORICAL    DEVELOPMENT    OF    PSEUDOSPHERICAL   SURFACES.  73 


tan  ^  =  c  ^''°^'' 
Li  the  second  expression  is  used  this  expression  becomes 

but  in  either  case,  as  Bianchi  said,  "The  initial  pseudospherical  surface  corre- 
sponding to  this  vahie  of  (o  is  Dini's  screw  surface,"  that  is,  the  surface  derived 
from  a  pseudospherical  helicoidal  surface  by  a  Bjicklund  transformation  is  a 
straight  line. 

From  the  study  of  individual  pseudospherical  surfaces,  Bianchi  *'■  *•  turned  to 
the  investigation  of  triply  orthogonal  systems  that  contain  a  family  of  surfaces 
with  constant  negative  curvature.  He  ''^^  then  proceeded  to  the  consideration  of 
such  a  system  as  a  unit  and  the  transformation  of  one  system  into  another  by  a 
complementary  or  Biicklund  transformation,  so  that  if  the  pseudospherical  sur- 
faces undergo  either  transformation  they  still  belong  to  a  triply  orthogonal  sys- 
tem. He  divided  these  systems  of  surfaces  into  two  classes  according  to  whether 
the  pseudospherical  surfaces  of  the  one  family  have  each  a  different  radius  of 
curvature  or  each  the  same  radius  of  curvature. 

Bianchi  did  not  devote  much  time  to  the  study  of  surfaces  of  the  first  class, 
but  wrote  several  papers  on  those  of  the  second  class,  which  he  called  a  Wein- 
garten  system  after  their  discoverer,  for  immediately  after  Bianchi  had  pub- 
lished his  proofs  of  Ribaucour's  theorems  on  the  cyclic  system  in  1884,  Wein- 
garten  wrote  to  him  suggesting  the  possibility  of  deriving  from  an  initial  surface 
of  constant  curvature  a  whole  family  of  surfaces  each  with  the  same  constant 
curvature  and  each  lying  at  an  infinitely  short  distance  from  the  one  next  to  it, 
while  the  family  to  which  they  belong  form  part  of  a  triply  orthogonal  system. 

In  this  way  the  transformation  theory  was  rapidly  but  continuously  developed 
without  any  overlapping  of  results  or  coincidence  of  discoveries.  Each  mathe- 
matician added  his  completed  proposition  to  the  theory  and  then  stood  aside 
while  the  next  one  took  it  up  and  developed  it  farther. 

Bianchi's  concejition  of  the  deformation  of  one  pseudospherical  surface  into 
others  of  the  same  kind  was  at  the  beginning  puiely  geometrical.  He  derived 
a  complementary  surface  by  cutting  off  equal  lengths  on  the  tangents  of  a  family 
of  geodesic  lines  going  out  fi-om  a  point  at  infinity  on  the  original  surface  ;  and 
his  analytic  work  simply  interpreted  this  geometrical  notion .  Next  Lie  showed 
that  the  geodesies,  consequently  the  new  surfaces,  are  obtainable  by  quadrature 
alone,  derived  oo"  surfaces  instead  of  oo'  from  the  given  one  and  wrote  down 
the  initial  equations  of  transformation  in  a  form  that  made  it  evident  they 
defined  equally  Ribaucour's  cyclic  system  and  Bianchi's  complementary  trans- 
formation.    Bianchi  proved  completely  the  identity  of  these  two  theorems  and 


74  E.    M.    CODDINGTON. 

in  demonstrating  Ribaucour's  propositions,  introduced  the  angle  6  that  a  radius 
of  a  circle  makes  with  a  line  of  curvature  of  an  orthogonal  surface  ;  Darboux 
made  use  of  this  angle  in  developing  a  new  set  of  equations  that  are  more  prac- 
tical for  actual  transformation  than  Lie's ;  Biicklund  introduced  a  new  trans- 
formation, a  generalization  of  Bianchi  in  which  the  tangent  planes  at  corre- 
sponding points  of  the  two  surfaces  meet  at  a  constant  angle  but  not  at  a  right 
angle.  Bianchi  gave  a  geometrical  method  for  performing  Bjicklund's  transfor- 
mation and  practically  completed  the  subject. 


TJIE  LffiRARY 
DIVERSITY  OF  CALIFORNIA 

LO&  ANGELES 


UNIVERSITY  OF  CALIFORNIA,  LOS  ANGELES 

THE  UNIVERSITY  LIBRARY 

This  book  is  DUE  on  the  last  date  stamped  below 


FEB  2  7  1969 


MAY  2  4  197) 
MAY  1  7  RttL' 


Form   L-ft 
25m-10.'tl(2lEilj 


*^  Coddington- 
645  -A  brief  ac- 
C64b  count   of  the 
historical 

development  of 
pseudoapherioal^suEs. 

faces. 


*QA 
645 
C64b 


tncmetHog  1 


111111111  llipilll  !ll|l!lll||lll| 


D     000  040  661      1 


M.72 


